#7644308

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

4740 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

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, 7551504);