#8430545

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

4972 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337741);