#96076

order	512
nilpotency class	3
isoclinism	isoclinic class of #95196
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,6,7,8)(9,10,12,17,14,18,13,16)(11,19,21,15,20,24,22,23)(25,26,29,33,30,34,28,32)(27,35,36,31,37,39,38,40);(1,2,4,6,5,7,3,8)(9,11,13,22,14,20,12,21)(10,15,17,24,18,23,16,19)(25,27)(26,31)(28,38)(29,36)(30,37)(32,35)(33,39)(34,40);(11,20)(15,23)(19,24)(21,22)(25,28,30,29)(26,32,34,33)(27,36,37,38)(31,39,40,35)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,d^2=gi,e^2=g,f^2=gh,b^a=bd,d^a=dg,f^a=fghi,c^b=cg,d^b=dh,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257622, #72257719
maximal normal subgroup of	#72257622, #72257719
maximal quotient of	#72265486

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3272);