#10102240

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4181 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10009436);