#7839624

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

4688 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7746820);