#8664354

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	#56299	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571550);