#8460693

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:

(1,2,8,11)(3,10,6,9)(4,7,5,15)(12,16,13,14)(17,18,23,27)(19,22,26,31)(20,30,29,24)(21,25,28,32)(33,34,40,38)(35,44,47,41)(36,48,42,37)(39,43,46,45);(1,3,8,6)(2,9,11,10)(4,12,5,13)(7,14,15,16)(17,19,23,26)(18,22,27,31)(20,21,29,28)(24,32,30,25)(33,35)(34,41)(36,45)(37,39)(38,44)(40,47)(42,43)(46,48);(1,4,8,5)(2,7,11,15)(3,12,6,13)(9,14,10,16)(17,20,23,29)(18,24,27,30)(19,28,26,21)(22,25,31,32)(33,36)(34,37)(35,45)(38,48)(39,41)(40,42)(43,47)(44,46);(1,5,8,4)(2,7,11,15)(3,13,6,12)(9,14,10,16)(17,19)(18,22)(20,28)(21,29)(23,26)(24,25)(27,31)(30,32)(33,37,40,48)(34,42,38,36)(35,46,47,39)(41,45,44,43);(1,6,8,3)(2,10,11,9)(4,13,5,12)(7,16,15,14)(17,21,23,28)(18,25,27,32)(19,29,26,20)(22,24,31,30)(33,38,40,34)(35,44,47,41)(36,48,42,37)(39,45,46,43);(1,7)(2,5)(3,14)(4,11)(6,16)(8,15)(9,13)(10,12)(17,22)(18,26)(19,27)(20,25)(21,24)(23,31)(28,30)(29,32)(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=gh,c^2=gh,d^2=g,e^2=gi,f^2=hi,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, 8367889);