#6358025

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358021
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=ei,d^c=dhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265221);