#8701304

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(19,21)(24,26)(27,28)(31,32);(2,3)(7,9)(8,10)(15,16)(17,18,23,22)(19,26,21,24)(20,25,29,30)(27,32,28,31)(33,34)(35,41)(36,37)(38,42)(39,45)(40,46)(43,47)(44,48);(17,19,23,21)(18,24,22,26)(20,27,29,28)(25,31,30,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20)(18,25)(19,27)(21,28)(22,30)(23,29)(24,31)(26,32)(33,36,38,46)(34,40,42,37)(35,43,45,48)(39,47,41,44);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,21,23,19)(18,24,22,26)(20,28,29,27)(25,31,30,32)(33,35)(34,41)(36,43)(37,47)(38,45)(39,42)(40,44)(46,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22,23,18)(19,26,21,24)(20,25,29,30)(27,31,28,32)(33,37,38,40)(34,36,42,46)(35,44,45,47)(39,43,41,48)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=gi,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, 8608500);