#10247492

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	#53438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4135 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=i,e^2=g,f^2=gh,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, 10154688);