#8772928

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	#55044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8680124);