#8521622

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	#37102	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8428818);