#495141

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1350 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=f,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224826, #72226816, #72226822, …
maximal normal subgroup of	#72224826, #72226816, #72226822, …
maximal quotient of	#72224822, #72319138, #72320142, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402337);