#7744649

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

5566 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=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, 7651845);