#6367723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1495 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274919);