#6811994

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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	C₂²×C₄	C₂⁵
prime radical	#3060	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6719190);