#8773004

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	#57368	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=h,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, 8680200);