#130285

order	512
nilpotency class	4
isoclinism	isoclinic class of #130284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8350
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#62275	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

601 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250025, #72250072, #72250223, …
maximal normal subgroup of	#72250025, #72250072, #72250223, …
maximal quotient of	#72250320, #72280533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37481);