#8992960

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	#55008	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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)(3,8)(4,10)(5,6)(7,12)(9,13)(11,16)(14,15)(17,18,22,21)(19,23,28,26)(20,24,29,30)(25,31,27,32)(34,36)(35,43)(38,48)(41,45);(1,3,7,14)(2,8,12,15)(4,11,13,5)(6,10,16,9)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,34,39,36)(35,44,43,40)(37,45,47,41)(38,42,48,46);(1,4,7,13)(2,9,12,10)(3,5,14,11)(6,8,16,15)(17,20,22,29)(18,24,21,30)(19,25,28,27)(23,31,26,32)(33,35)(34,40)(36,44)(37,38)(39,43)(41,42)(45,46)(47,48);(1,4)(2,10)(3,11)(5,14)(6,15)(7,13)(8,16)(9,12)(17,21,22,18)(19,26,28,23)(20,24,29,30)(25,31,27,32)(33,36,39,34)(35,44,43,40)(37,45,47,41)(38,46,48,42);(1,5,7,11)(2,6,12,16)(3,13,14,4)(8,9,15,10)(17,20)(18,24)(19,27)(21,30)(22,29)(23,32)(25,28)(26,31)(33,37)(34,41)(35,38)(36,45)(39,47)(40,42)(43,48)(44,46);(1,6)(2,11)(3,10)(4,15)(5,12)(7,16)(8,13)(9,14)(17,19)(18,23)(20,27)(21,26)(22,28)(24,32)(25,29)(30,31)(33,38)(34,42)(35,37)(36,46)(39,48)(40,41)(43,47)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=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, 8900156);