#8093679

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

5092 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000875);