#7680026

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

4705 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7587222);