#7693696

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4662 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=gi,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, 7600892);