#7521373

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 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)(13,14,19,25)(15,27,30,21)(16,31,28,42)(17,35,36,41)(18,26,20,24)(22,33,43,34)(23,38,39,32)(29,40,37,44);(1,2)(3,4)(6,10)(11,12)(13,15,19,30)(14,21,25,27)(16,32,28,38)(17,33,36,34)(18,29,20,37)(22,41,43,35)(23,31,39,42)(24,40,26,44);(13,16,19,28)(14,22,25,43)(15,17,30,36)(18,34,20,33)(21,39,27,23)(24,31,26,42)(29,38,37,32)(35,44,41,40);(2,4)(5,7,9,8)(6,11,10,12)(13,17)(14,23)(15,28)(16,30)(18,32)(19,36)(20,38)(21,22)(24,35)(25,39)(26,41)(27,43)(29,34)(31,44)(33,37)(40,42);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,29,30,37)(16,33,28,34)(17,32,36,38)(21,40,27,44)(22,42,43,31)(23,35,39,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^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,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7428569);