#7481532

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7388728);