#8456008

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

4199 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=g,e^2=gi,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, 8363204);