#10320953

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

4736 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,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, 10228149);