#358311

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265507);