#7309103

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
prime radical	#42432	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1644 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)(3,7)(4,6)(5,8)(17,18,22,24,21,25,20,26)(19,23,29,32,27,31,28,30);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,27)(18,23,25,31)(20,28,22,29)(24,32,26,30);(9,11)(10,13)(12,16)(14,15)(18,24)(19,27)(20,22)(23,30)(25,26)(31,32);(3,8)(5,7)(10,14)(11,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,32)(25,31)(26,30);(2,6)(5,7)(10,14)(13,15)(18,25)(19,27)(24,26)(28,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ch,d^b=dfh,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, 7216299);