#8096989

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(3,5,12,9)(4,13,6,10)(11,16,14,15)(18,24)(20,25)(26,32)(27,30)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,9)(4,14)(5,7)(6,11)(8,12)(10,15)(13,16)(17,18,23,24)(19,25,29,20)(21,26,28,32)(22,27,31,30)(35,42)(39,46)(43,47)(44,48);(1,4,8,6)(2,10,7,13)(3,11,12,14)(5,15,9,16)(17,19,23,29)(18,25,24,20)(21,22,28,31)(26,27,32,30)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,5)(2,3)(4,15)(6,16)(7,12)(8,9)(10,11)(13,14)(17,20)(18,19)(21,30)(22,26)(23,25)(24,29)(27,28)(31,32);(1,6,8,4)(2,10,7,13)(3,11,12,14)(5,16,9,15)(17,21)(18,26)(19,22)(20,30)(23,28)(24,32)(25,27)(29,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,7,8,2)(3,5,12,9)(4,13,6,10)(11,15,14,16)(17,22)(18,27)(19,28)(20,26)(21,29)(23,31)(24,30)(25,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004185);