#6490781

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=ghi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6397977);