#7515014

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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)(7,10)(8,9)(11,12)(13,14,20,15)(16,27,30,28)(17,33,36,38)(18,24,19,23)(21,32,25,31)(22,35,26,34)(29,37,42,39)(40,44,41,43);(1,2)(3,4)(6,9)(10,12)(13,15,20,14)(16,28,30,27)(17,33,36,38)(18,23,19,24)(21,31,25,32)(22,35,26,34)(29,37,42,39)(40,44,41,43);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,29)(18,32)(19,31)(20,30)(22,37)(23,28)(24,27)(26,39)(33,40)(34,44)(35,43)(36,42)(38,41);(2,4)(13,17,20,36)(14,22,15,26)(16,29,30,42)(18,35,19,34)(21,37,25,39)(23,38,24,33)(27,40,28,41)(31,44,32,43);(6,9)(10,12)(16,30)(21,25)(27,28)(29,42)(31,32)(37,39)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,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, 7422210);