#8696238

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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)(3,9)(4,5)(6,10)(7,8)(11,14)(12,13)(15,16)(17,18,23,27)(19,24,28,30)(20,25,29,32)(21,22,26,31)(34,39)(35,42)(36,44)(45,47);(2,8)(4,9)(10,16)(12,14)(19,28)(20,29)(24,30)(25,32)(33,34,38,39)(35,37,42,40)(36,41,44,46)(43,45,48,47);(1,3,7,5)(2,9,8,4)(6,11,15,13)(10,14,16,12)(17,19,23,28)(18,24,27,30)(20,22,29,31)(21,25,26,32)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,4)(2,3)(5,8)(6,14)(7,9)(10,13)(11,16)(12,15)(17,20)(18,25)(19,22)(21,30)(23,29)(24,26)(27,32)(28,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,5,7,3)(2,4,8,9)(6,11,15,13)(10,14,16,12)(17,21)(18,22)(19,25)(20,30)(23,26)(24,29)(27,31)(28,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,6,7,15)(2,10,8,16)(3,11,5,13)(4,12,9,14)(17,22)(18,26)(19,29)(20,28)(21,27)(23,31)(24,32)(25,30)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,e^2=gh,f^2=g,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, 8603434);