#6672306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667760
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	#3013	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

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

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, 6579502);