#8188517

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,8,7)(3,12,6,9)(4,11,5,10)(13,15,14,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34,37,40)(35,38,43,41)(36,44,45,39)(42,47,48,46);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=ghi,e^2=h,f^2=g,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, 8095713);