#376624

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283820);