#7517832

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 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)(7,12)(8,10)(9,11)(13,14,20,26)(15,21,28,39)(16,31,27,42)(17,35,36,44)(18,25,19,24)(22,34,38,33)(23,37,43,32)(29,41,30,40);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,21)(16,32)(17,33)(18,30)(19,29)(20,28)(22,35)(23,31)(24,41)(25,40)(26,39)(27,37)(34,36)(38,44)(42,43);(13,16,20,27)(14,22,26,38)(15,17,28,36)(18,34,19,33)(21,23,39,43)(24,42,25,31)(29,37,30,32)(35,40,44,41);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,27)(16,28)(18,37)(19,32)(20,36)(21,38)(22,39)(24,44)(25,35)(26,43)(29,33)(30,34)(31,40)(41,42);(6,11)(10,12)(15,28)(17,36)(21,39)(23,43)(29,30)(32,37)(35,44)(40,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,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,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 7425028);