#10247266

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	#56587	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3843 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154462);