#8696084

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	#50323	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,37,40)(35,41,43,38)(36,39,46,45)(42,48,44,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(18,19)(23,25)(24,26)(30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,48)(45,46);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,48)(45,46);(1,4,7,2)(3,8,5,10)(6,9,14,12)(11,16,13,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35,37,43)(34,38,40,41)(36,44,46,42)(39,48,45,47);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,36,37,46)(34,39,40,45)(35,42,43,44)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8603280);