#7673863

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4418 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581059);