#10320801

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

4956 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)(11,12)(13,16)(14,15)(17,18,22,23)(19,20,26,27)(21,24,29,30)(25,28,31,32);(1,2)(3,7)(4,5)(6,8)(10,15)(11,13)(18,23)(20,27)(21,29)(25,31);(2,5)(6,7)(10,15)(12,16)(17,19,22,26)(18,20,23,27)(21,25,29,31)(24,28,30,32);(2,5)(6,7)(9,11,14,13)(10,16,15,12)(17,20,22,27)(18,19,23,26)(21,28,29,32)(24,25,30,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,12,15,16)(17,21,22,29)(18,24,23,30)(19,25,26,31)(20,28,27,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,f^2=g,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, 10227997);