#6380414

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287610);