#7993277

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

4509 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7900473);