#7564108

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471304);