#7847238

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

4991 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=i,f^2=gi,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, 7754434);