#10363667

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	#53423	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,18)(19,27)(20,29)(21,26)(22,24)(23,30)(25,31)(28,32)(33,34,39,42)(35,40,36,38)(37,41,46,48)(43,47,44,45);(2,6)(3,10)(8,14)(11,13)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(35,36)(38,40)(43,44)(45,47);(4,12)(8,14)(11,13)(15,16)(18,24)(21,32)(23,27)(25,29)(33,35,39,36)(34,40,42,38)(37,44,46,43)(41,45,48,47);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,20,22,31)(18,25,24,29)(19,28,30,26)(21,23,32,27)(33,36,39,35)(34,38,42,40)(37,43,46,44)(41,47,48,45);(1,4,5,12)(2,8,6,14)(3,11,10,13)(7,15,9,16)(17,21,22,32)(18,26,24,28)(19,29,30,25)(20,23,31,27)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,38)(34,35)(36,42)(37,45)(39,40)(41,43)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10270863);