#8751917

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	#53608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 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:

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

Polycyclic presentation:

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

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, 8659113);