#10360904

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10268100);