#9206598

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	#53445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 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)(3,12)(4,9)(5,10)(6,8)(7,11)(13,14)(15,16)(17,18)(19,26)(20,30)(21,24)(22,25)(23,29)(27,28)(31,32)(33,34,39,44)(35,42,37,40)(36,38,43,41)(45,48,47,46);(1,3,7,6)(2,8,11,12)(4,13,10,15)(5,14,9,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,35,39,37)(34,40,44,42)(36,47,43,45)(38,48,41,46);(1,4,7,10)(2,9,11,5)(3,13,6,15)(8,16,12,14)(17,20,22,24)(18,21,25,30)(19,27,29,31)(23,28,26,32)(33,36,39,43)(34,41,44,38)(35,45,37,47)(40,48,42,46);(1,4)(2,9)(3,13)(5,11)(6,15)(7,10)(8,16)(12,14)(17,21,22,30)(18,24,25,20)(19,28,29,32)(23,31,26,27)(33,37,39,35)(34,42,44,40)(36,47,43,45)(38,48,41,46);(1,5,7,9)(2,10,11,4)(3,14,6,16)(8,15,12,13)(17,18)(19,26)(20,21)(22,25)(23,29)(24,30)(27,32)(28,31)(33,38,39,41)(34,43,44,36)(35,46,37,48)(40,47,42,45);(1,6,7,3)(2,8,11,12)(4,15,10,13)(5,14,9,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113794);