#6672632

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
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

1603 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)(19,28)(20,25)(21,36)(22,26)(23,27)(24,30)(29,43)(31,39)(32,42)(33,35)(34,40)(37,41)(38,45)(44,48)(46,47);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,18)(19,28)(20,33)(21,26)(22,36)(23,27)(24,31)(25,35)(29,38)(30,39)(32,42)(34,41)(37,40)(43,45)(44,47)(46,48);(17,19,23,32)(18,24,27,39)(20,29,34,44)(21,31,22,30)(25,38,40,47)(26,42,36,28)(33,43,41,48)(35,46,37,45);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(20,34)(25,40)(29,44)(33,41)(35,37)(38,47)(43,48)(45,46);(3,11)(6,9)(10,16)(14,15)(17,20)(18,25)(19,29)(21,35)(22,37)(23,34)(24,38)(26,41)(27,40)(28,43)(30,45)(31,46)(32,44)(33,36)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=f,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,h^c=hi,e^d=ei,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, 6579828);