#7565182

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1822 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|c^2=fgi,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472378);