#10397295

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

5139 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10304491);