#7671257

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

4814 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)(3,4)(5,8)(6,7)(10,14)(15,16)(18,23)(21,29)(24,30)(27,32);(1,3,6,5)(2,4,7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,23)(19,24,28,30)(20,21,25,29)(26,27,31,32);(2,7)(4,8)(18,23)(20,25)(24,30)(26,31);(1,4,6,8)(2,3,7,5)(9,11,13,12)(10,15,14,16)(17,19,22,28)(18,24,23,30)(20,26,25,31)(21,27,29,32);(1,3,6,5)(2,4,7,8)(9,12,13,11)(10,16,14,15)(17,20,22,25)(18,21,23,29)(19,26,28,31)(24,27,30,32);(1,5,6,3)(2,8,7,4)(17,21)(18,25)(19,27)(20,23)(22,29)(24,31)(26,30)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578453);