#9835839

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3737 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=ghi,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, 9743035);