#7837852

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

4940 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^2=gh,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, 7745048);