#8202290

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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)(3,10)(4,12)(5,7)(6,8)(9,14)(11,15)(13,16)(17,18,23,22)(19,26,24,20)(21,29,30,25)(27,28,31,32)(33,34)(35,37)(36,44)(38,42)(39,40)(41,47)(43,46)(45,48);(1,3,7,6)(2,8,5,10)(4,13,14,11)(9,16,12,15)(17,19,23,24)(18,20,22,26)(21,27,30,31)(25,28,29,32)(34,40)(35,42)(41,44)(43,48);(1,2,7,5)(3,8,6,10)(4,9,14,12)(11,15,13,16)(17,20,23,26)(18,24,22,19)(21,28,30,32)(25,31,29,27)(33,35,39,42)(34,37,40,38)(36,43,47,48)(41,45,44,46);(1,4)(2,9)(3,11)(5,12)(6,13)(7,14)(8,15)(10,16)(17,19)(18,20)(21,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,36,39,47)(34,41,40,44)(35,43,42,48)(37,45,38,46);(1,5,7,2)(3,8,6,10)(4,12,14,9)(11,15,13,16)(17,21,23,30)(18,25,22,29)(19,27,24,31)(20,28,26,32)(33,37,39,38)(34,35,40,42)(36,45,47,46)(41,43,44,48);(1,6,7,3)(2,8,5,10)(4,13,14,11)(9,15,12,16)(17,22,23,18)(19,26,24,20)(21,29,30,25)(27,32,31,28)(33,38,39,37)(34,42,40,35)(36,46,47,45)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=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, 8109486);