#10440027

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,8)(4,10)(5,7)(6,9)(11,15)(12,16)(13,14)(18,19)(20,27)(21,29)(30,31)(33,34,39,42)(35,40,37,45)(36,38,41,48)(43,44,46,47);(2,5)(8,10)(9,13)(15,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(35,37)(40,45)(43,46)(44,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35,39,37)(34,40,42,45)(36,43,41,46)(38,44,48,47);(1,4,7,3)(2,8,5,10)(6,11,14,12)(9,16,13,15)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36)(34,38)(35,43)(37,46)(39,41)(40,44)(42,48)(45,47);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,35)(34,40,42,45)(36,43,41,46)(38,47,48,44);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,38)(34,41)(35,44)(36,42)(37,47)(39,48)(40,46)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10347223);