#8534888

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442084);