#8467219

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

3931 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)(3,8,13,11)(4,9,14,16)(5,15,12,10)(17,18,24,22)(19,28,31,25)(20,32,26,21)(23,27,30,29)(34,39)(36,44)(40,42)(43,45);(1,2,7,6)(3,11,13,8)(4,9,14,16)(5,10,12,15)(17,19)(18,25)(20,29)(21,23)(22,28)(24,31)(26,27)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,3,7,13)(2,8,6,11)(4,12,14,5)(9,10,16,15)(17,20)(18,21)(19,29)(22,32)(23,25)(24,26)(27,31)(28,30)(33,35,38,37)(34,40,39,42)(36,43,44,45)(41,47,46,48);(1,4)(2,9)(3,12)(5,13)(6,16)(7,14)(8,10)(11,15)(17,21,24,32)(18,26,22,20)(19,30,31,23)(25,29,28,27)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,5,7,12)(2,10,6,15)(3,4,13,14)(8,16,11,9)(17,22,24,18)(19,28,31,25)(20,32,26,21)(23,29,30,27)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6,7,2)(3,11,13,8)(4,9,14,16)(5,15,12,10)(17,23,24,30)(18,27,22,29)(19,21,31,32)(20,25,26,28)(33,37,38,35)(34,42,39,40)(36,43,44,45)(41,47,46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8374415);