#8587653

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	#36909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,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,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494849);