#8436235

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

4022 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)(19,27)(20,26)(21,23)(22,25)(24,28)(29,31)(30,32)(33,34,40,38)(35,44,47,41)(36,48,42,37)(39,43,46,45);(1,3,8,6)(2,9,7,12)(4,14,5,13)(10,16,11,15)(17,19,24,22)(18,25,28,27)(20,29,21,30)(23,31,26,32)(33,35)(34,41)(36,45)(37,39)(38,44)(40,47)(42,43)(46,48);(1,4,8,5)(2,10,7,11)(3,13,6,14)(9,15,12,16)(17,20,24,21)(18,26,28,23)(19,29,22,30)(25,32,27,31)(33,36)(34,37)(35,45)(38,48)(39,41)(40,42)(43,47)(44,46);(1,5,8,4)(2,11,7,10)(3,14,6,13)(9,16,12,15)(17,21,24,20)(18,26,28,23)(19,30,22,29)(25,32,27,31)(33,37,40,48)(34,42,38,36)(35,46,47,39)(41,45,44,43);(1,6,8,3)(2,12,7,9)(4,14,5,13)(10,16,11,15)(17,22,24,19)(18,27,28,25)(20,30,21,29)(23,32,26,31)(33,38,40,34)(35,44,47,41)(36,48,42,37)(39,45,46,43);(1,7,8,2)(3,12,6,9)(4,11,5,10)(13,16,14,15)(17,23,24,26)(18,21,28,20)(19,31,22,32)(25,30,27,29)(33,39,40,46)(34,43,38,45)(35,37,47,48)(36,41,42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=g,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, 8343431);