#7308718

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#31017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7215914);