#8696678

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

4111 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,31)(28,29)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,36,38,35)(34,40,42,39)(37,43,46,44)(41,47,45,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603874);