#9794303

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	#51957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9701499);