#7484512

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309080
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1491 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ch,d^b=dfh,e^b=ehi,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, 7391708);