#7691824

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

4799 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^2=g,f^2=g,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, 7599020);