#232081

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139277);