#137398

order	512
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(5,6,10,15,11,17,9,16,12,19,8,26,29,18,27,13)(7,20,23,14,24,31,22,32,25,36,21,34,30,35,28,33);(1,2,3,4)(6,13,19,16)(8,27)(9,10)(11,29)(14,31,34,35)(15,17,26,18)(20,33,36,32)(21,28)(22,23)(24,30);(5,7,12,25)(6,14,19,34)(8,28,10,22)(9,23,27,21)(11,30,29,24)(13,31,16,35)(15,20,26,36)(17,33,18,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=g,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,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	#72250968, #72250970, #72257006, …
maximal normal subgroup of	#72250968, #72250970, #72257006, …
maximal quotient of	#72265567

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44594);