#7517043

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	#24455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2012 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)(13,14,20,24)(15,21,29,40)(16,30,34,36)(17,33,22,32)(18,23,19,37)(25,41,44,35)(26,31,38,43)(27,39,28,42);(1,2)(3,4)(6,9)(10,12)(13,15,20,29)(14,21,24,40)(16,31,34,43)(17,35,22,41)(18,28,19,27)(23,42,37,39)(25,32,44,33)(26,30,38,36);(13,16,20,34)(14,22,24,17)(15,25,29,44)(18,33,19,32)(21,38,40,26)(23,36,37,30)(27,31,28,43)(35,42,41,39);(2,4)(5,7,8,11)(6,10,9,12)(13,17,20,22)(14,16,24,34)(15,26,29,38)(18,36,19,30)(21,25,40,44)(23,32,37,33)(27,35,28,41)(31,39,43,42);(6,9)(10,12)(13,18,20,19)(14,23,24,37)(15,27,29,28)(16,32,34,33)(17,36,22,30)(21,39,40,42)(25,43,44,31)(26,35,38,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fi,e^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,e^b=ei,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7424239);