#9794308

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	#55163	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,22,24)(19,27,29,23)(20,25,28,32)(21,30,31,26)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,22,29)(18,23,24,27)(20,21,28,31)(25,26,32,30)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,7)(3,10)(8,15)(11,12)(18,24)(23,27)(25,32)(26,30)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,16)(10,12)(17,20)(18,25)(19,21)(22,28)(23,26)(24,32)(27,30)(29,31)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)(33,39,40,47)(34,42,37,45)(35,43,41,48)(36,44,38,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=gh,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, 9701504);