#7678660

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

4980 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585856);