#7812000

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

5092 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,e^2=g,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, 7719196);