#7683121

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

4721 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7590317);