#7718110

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

4839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7625306);