#7674160

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4529 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)(9,10)(11,14)(12,16)(13,15)(17,18,24,26)(19,27,22,21)(20,25,23,32)(28,31,29,30);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,15,12)(17,19,24,22)(18,21,26,27)(20,29,23,28)(25,31,32,30);(9,12)(10,11)(13,14)(15,16)(17,20,24,23)(18,25,26,32)(19,28,22,29)(21,30,27,31);(9,10)(11,14)(12,16)(13,15)(17,21)(18,19)(20,30)(22,26)(23,31)(24,27)(25,28)(29,32);(17,22,24,19)(18,21,26,27)(20,28,23,29)(25,31,32,30);(1,4,5,2)(3,7,8,6)(17,23,24,20)(18,25,26,32)(19,28,22,29)(21,31,27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,e^2=i,f^2=g,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, 7581356);