#9814823

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	#53416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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)(3,10)(4,12)(5,8)(6,9)(7,14)(11,13)(15,16)(17,18,22,20)(19,23,25,26)(21,24,30,28)(27,31,29,32)(33,34)(35,43)(36,46)(37,40)(38,42)(39,41)(44,47)(45,48);(1,3,6,8)(2,5,9,10)(4,11,14,15)(7,13,12,16)(19,25)(23,26)(27,29)(31,32)(33,35,40,39)(34,41,37,43)(36,45,38,44)(42,48,46,47);(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36,40,38)(34,42,37,46)(35,44,39,45)(41,47,43,48);(1,4)(2,7)(3,11)(5,13)(6,14)(8,15)(9,12)(10,16)(17,20,22,18)(19,26,25,23)(21,28,30,24)(27,32,29,31)(33,37,40,34)(35,43,39,41)(36,42,38,46)(44,47,45,48);(1,5,6,10)(2,8,9,3)(4,13,14,16)(7,15,12,11)(17,19)(18,23)(20,26)(21,29)(22,25)(24,32)(27,30)(28,31)(33,38,40,36)(34,42,37,46)(35,45,39,44)(41,47,43,48);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)(33,39,40,35)(34,43,37,41)(36,45,38,44)(42,48,46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=i,e^2=g,f^2=h,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, 9722019);