#8696724

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	#56325	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4141 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18)(19,24)(20,26)(21,30)(22,28)(23,29)(25,31)(27,32)(33,34,37,40)(35,41,43,38)(36,39,46,45)(42,48,44,47);(2,3)(7,9)(8,10)(14,15)(17,19,23,22)(18,24,29,28)(20,27,31,30)(21,26,32,25)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,48)(45,46);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,20)(18,25)(19,30)(21,28)(22,27)(23,31)(24,32)(26,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,48)(45,46);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,26)(19,27)(21,28)(22,30)(23,31)(24,32)(25,29)(33,35,37,43)(34,38,40,41)(36,44,46,42)(39,48,45,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,27)(19,26)(20,28)(22,25)(23,32)(24,31)(29,30)(33,34)(35,38)(36,45)(37,40)(39,46)(41,43)(42,48)(44,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,19)(18,28,29,24)(20,27,31,30)(21,26,32,25)(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=ghi,d^2=g,f^2=h,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, 8603920);