#8528962

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(3,4)(5,12)(6,9)(7,8)(10,11)(13,14)(15,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(34,37)(35,36)(41,45)(42,44);(1,3,7,5)(2,4,8,12)(6,10,15,13)(9,11,16,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(2,8)(3,5)(9,16)(10,13)(21,29)(24,31)(26,32)(28,30)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4)(2,3)(5,8)(6,14)(7,12)(9,13)(10,16)(11,15)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,5,7,3)(2,4,8,12)(6,13,15,10)(9,11,16,14)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,6)(2,9)(3,10)(4,11)(5,13)(7,15)(8,16)(12,14)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=g,e^2=h,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, 8436158);