#7515254

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

1956 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)(17,31)(19,24)(20,28)(21,32)(26,33)(30,34)(36,39)(38,40)(41,43)(42,44);(1,2)(3,4)(6,9)(10,12)(13,14,18,22)(15,24,26,39)(16,28,30,40)(17,32,31,21)(19,33,36,23)(20,34,38,27)(25,35,29,37)(41,44,43,42);(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,29)(17,33)(18,26)(20,37)(21,39)(22,36)(23,31)(24,32)(25,30)(27,43)(28,44)(34,41)(35,38)(40,42);(2,4)(13,16,18,30)(14,20,22,38)(15,25,26,29)(17,34,31,27)(19,35,36,37)(21,40,32,28)(23,41,33,43)(24,42,39,44);(6,9)(10,12)(15,26)(19,36)(23,33)(24,39)(25,29)(35,37)(41,43)(42,44)

Polycyclic presentation:

2^9|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,d^c=di,e^c=ei,g^c=gi,h^c=hi,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, 7422450);