#7684835

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4571 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)(5,8)(6,7)(10,13)(14,16)(17,18)(19,22)(20,21)(23,25)(24,26)(27,30)(28,29)(31,32);(9,10,12,13)(11,14,15,16)(17,19,24,21)(18,22,26,20)(23,27,31,29)(25,30,32,28);(2,7)(3,8)(17,20,24,22)(18,19,26,21)(23,28,31,30)(25,27,32,29);(1,3,6,8)(2,4,7,5)(9,11,12,15)(10,14,13,16)(17,21,24,19)(18,20,26,22)(23,29,31,27)(25,28,32,30);(1,4,6,5)(2,3,7,8)(17,22,24,20)(18,19,26,21)(23,28,31,30)(25,29,32,27);(1,5,6,4)(2,8,7,3)(17,23,24,31)(18,25,26,32)(19,27,21,29)(20,28,22,30)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=h,f^2=h,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, 7592031);