#7671256

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

4804 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)(9,10,13,12)(11,15,16,14)(17,18,23,21)(19,28,30,24)(20,26,31,25)(22,32,29,27);(1,3,6,5)(2,4,7,8)(9,11,13,16)(10,14,12,15)(17,19,23,30)(18,24,21,28)(20,29,31,22)(25,27,26,32);(2,7)(4,8)(9,10,13,12)(11,14,16,15)(17,20,23,31)(18,25,21,26)(19,22,30,29)(24,32,28,27);(1,4,6,8)(2,3,7,5)(17,21,23,18)(19,28,30,24)(20,25,31,26)(22,32,29,27);(1,3,6,5)(2,4,7,8)(9,12,13,10)(11,15,16,14)(17,20)(18,26)(19,29)(21,25)(22,30)(23,31)(24,32)(27,28);(1,5,6,3)(2,8,7,4)(17,22,23,29)(18,27,21,32)(19,20,30,31)(24,26,28,25)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578452);