#9224300

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3883 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9131496);