#377671

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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)(11,21,24,32,25,31,23,16)(12,26,29,33,27,34,28,17)(22,36,39,40,37,30,38,35);(1,2)(3,4)(9,11,13,23,15,25,14,24)(10,16,19,21,20,32,18,31)(12,22,28,38,27,37,29,39)(17,30,26,35,33,36,34,40);(2,4)(6,8)(9,12,13,28,15,27,14,29)(10,17,19,26,20,33,18,34)(11,22,23,38,25,37,24,39)(16,30,21,35,32,36,31,40);(10,18,20,19)(12,27)(16,31,32,21)(17,26,33,34)(22,37)(28,29)(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^c=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, 284867);