#8679886

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3933 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,22,26)(19,23,20,24)(21,25,30,29)(27,31,28,32)(33,34)(35,41)(36,46)(37,43)(38,39)(40,44)(42,45)(47,48);(2,4)(7,9)(8,12)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,32)(29,30)(33,35,40,38)(34,41,44,39)(36,47,45,37)(42,43,46,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,36)(34,42)(35,37)(38,47)(39,43)(40,45)(41,48)(44,46);(1,4,6,2)(3,9,11,7)(5,8,14,12)(10,15,13,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,37,40,47)(34,43,44,48)(35,45,38,36)(39,46,41,42);(1,3)(2,7)(4,9)(5,13)(6,11)(8,16)(10,14)(12,15)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,38,40,35)(34,41,44,39)(36,47,45,37)(42,48,46,43);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,39,40,41)(34,38,44,35)(36,43,45,48)(37,42,47,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=ghi,d^2=h,e^2=gh,f^2=gi,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, 8587082);