#10412048

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	#37527	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10319244);