#7846753

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

4874 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,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, 7753949);