#137391

order	512
nilpotency class	4
isoclinism	isoclinic class of #12114
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9041
prime radical	#33530	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

564 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,13,32,47,8,15,33,46,31,14,5,11)(3,16,20,40,21,45,18,39,22,50,29,9,34,37,19,38)(4,23,26,44,27,59,24,43,28,48,30,10,35,41,25,42)(17,36,52,58,53,61,56,64,54,63,55,49,57,62,51,60);(1,3,7,34,8,22,31,21)(2,9,13,39,15,40,14,38)(4,17,35,53,28,54,27,57)(5,29,6,18,32,20,33,19)(10,36,43,61,44,63,42,62)(11,45,12,16,47,37,46,50)(23,58,41,64,48,49,59,60)(24,52,26,56,25,55,30,51);(1,4)(2,10)(3,17)(5,30)(6,24)(7,35)(8,28)(9,36)(11,41)(12,48)(13,43)(14,42)(15,44)(16,49)(18,52)(19,55)(20,56)(21,57)(22,54)(23,46)(25,33)(26,32)(27,31)(29,51)(34,53)(37,60)(38,62)(39,61)(40,63)(45,64)(47,59)(50,58)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=egh,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fh,g^c=gi,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217918
maximal normal subgroup of	#72217918
maximal quotient of	#72216736, #72216744, #72216747, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44587);