#7516608

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

2044 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)(15,23)(16,27)(17,30)(19,24)(20,28)(21,31)(26,32)(29,33)(35,37)(36,38)(39,42)(40,44);(1,2)(3,4)(6,9)(10,12)(13,14,18,22)(15,24,26,37)(16,28,29,38)(17,31,30,21)(19,32,35,23)(20,33,36,27)(25,34,41,43)(39,44,42,40);(13,15,18,26)(14,19,22,35)(16,25,29,41)(17,32,30,23)(20,34,36,43)(21,37,31,24)(27,39,33,42)(28,40,38,44);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,25)(17,33)(18,29)(19,34)(21,38)(22,36)(23,39)(24,40)(26,41)(27,30)(28,31)(32,42)(35,43)(37,44);(6,9)(10,12)(16,29)(20,36)(25,41)(27,33)(28,38)(34,43)(39,42)(40,44)

Polycyclic presentation:

2^9|b^2=i,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,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, 7423804);