#7682537

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) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,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, 7589733);