#10440013

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	#50837	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=g,e^2=i,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, 10347209);