#183471

order	512
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=be,d^b=di,e^b=eh,f^b=fh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320172, #72320205, #72320225
maximal normal subgroup of	#72320172, #72320205, #72320225
maximal quotient of	#72320434

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90667);