#8097748

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

5465 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 8004944);