#377672

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

975 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:

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

Polycyclic presentation:

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

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, 284868);