#7720342

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

4845 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)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,26,25,22)(21,28,30,24)(27,31,29,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,26)(21,29)(23,25)(24,32)(27,30)(28,31);(2,7)(4,6)(18,20)(19,25)(21,30)(31,32);(1,4,5,6)(2,3,7,8)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,29,31);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,27)(20,28)(22,31)(23,30)(25,29)(26,32);(17,22,23,26)(18,25,20,19)(21,31,30,32)(24,29,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627538);