#7715216

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

5158 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622412);