#8649030

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

3877 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=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, 8556226);