#7651236

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

5174 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7558432);