#7623500

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
prime radical	#33752	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1336 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)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,25)(21,24)(22,35)(23,29)(26,34)(27,33)(30,36)(31,38)(32,40)(37,46)(39,44)(41,48)(42,47)(43,45);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,29)(21,30)(22,31)(24,39)(25,28)(26,37)(27,38)(32,44)(33,45)(34,47)(35,43)(36,40)(41,42)(46,48);(3,10)(6,9)(11,16)(13,15)(18,24)(19,29)(21,34)(22,33)(23,36)(25,40)(26,27)(28,39)(30,46)(32,48)(35,41)(37,43)(38,42)(44,47);(2,7)(6,9)(8,14)(13,15)(18,25)(23,28)(24,40)(26,41)(27,35)(36,39)(37,42)(38,43);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)

Polycyclic presentation:

2^9|e^2=fi,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530696);