#8197077

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

5305 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)(3,5)(4,6)(7,8)(17,18,23,25)(19,26,20,22)(21,29,31,24)(27,30,28,32);(2,3)(6,7)(10,11)(14,15)(17,18,23,25)(19,26,20,22)(21,24,31,29)(27,32,28,30);(4,8)(6,7)(9,10,13,11)(12,15,16,14)(17,19,23,20)(18,22,25,26)(21,27,31,28)(24,30,29,32);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,23,19)(18,22,25,26)(21,28,31,27)(24,30,29,32);(17,21,23,31)(18,24,25,29)(19,27,20,28)(22,30,26,32);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,11,15)(17,22)(18,19)(20,25)(21,30)(23,26)(24,27)(28,29)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104273);