#9863410

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	#55579	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,24)(19,27,29,23)(20,25,28,32)(21,30,31,26)(33,34,39,44)(35,40,46,37)(36,42,48,41)(38,47,45,43);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,29)(18,23,24,27)(20,21,28,31)(25,26,32,30)(33,35,39,46)(34,40,44,37)(36,45,48,38)(41,47,42,43);(2,3)(7,9)(8,10)(14,15)(18,24)(23,27)(25,32)(26,30)(33,36,39,48)(34,41,44,42)(35,38,46,45)(37,47,40,43);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,32)(30,31)(33,36)(34,42)(35,45)(37,47)(38,46)(39,48)(40,43)(41,44);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,25)(19,21)(22,28)(23,26)(24,32)(27,30)(29,31)(33,37)(34,35)(36,47)(38,41)(39,40)(42,45)(43,48)(44,46);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,26)(19,28)(20,29)(22,31)(23,32)(24,30)(25,27)(33,38,39,45)(34,43,44,47)(35,36,46,48)(37,42,40,41)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=ghi,e^2=ghi,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, 9770606);