#8634462

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

3915 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,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, 8541658);