#7685645

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592841);