#10156453

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

5759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10063649);