#8097178

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

5977 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004374);