#10195863

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	#54608	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=gh,e^2=gh,f^2=ghi,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, 10103059);