#7682068

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

4963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589264);