#10188272

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	#49682	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3930 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)(3,9)(4,5)(6,10)(7,13)(8,14)(11,15)(12,16)(17,18,22,25)(19,23,29,32)(20,24,30,21)(26,28,27,31)(33,34,39,44)(35,43,38,40)(36,37,42,41)(45,46,47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,32)(26,30)(33,35,39,38)(34,40,44,43)(36,45,42,47)(37,48,41,46);(17,20)(18,24)(19,26)(21,25)(22,30)(23,28)(27,29)(31,32)(33,36,39,42)(34,41,44,37)(35,45,38,47)(40,46,43,48);(1,4,6,14)(2,8,10,5)(3,11,13,16)(7,15,9,12)(17,20)(18,24)(19,27)(21,25)(22,30)(23,31)(26,29)(28,32)(33,36)(34,41)(35,45)(37,44)(38,47)(39,42)(40,46)(43,48);(1,5,6,8)(2,4,10,14)(3,12,13,15)(7,11,9,16)(17,19)(18,23)(20,26)(21,31)(22,29)(24,28)(25,32)(27,30)(33,37,39,41)(34,42,44,36)(35,46,38,48)(40,45,43,47);(1,3)(2,9)(4,11)(5,15)(6,13)(7,10)(8,12)(14,16)(17,21,22,24)(18,20,25,30)(19,28,29,31)(23,27,32,26)(33,38,39,35)(34,43,44,40)(36,47,42,45)(37,46,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10095468);