#9203767

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	#40359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=gi,f^2=g,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, 9110963);