#7516752

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×#3269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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)(13,14,19,24)(15,20,26,39)(16,28,25,41)(17,31,33,44)(18,34,37,23)(21,32,38,30)(22,36,42,29)(27,43,35,40);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,20,24,39)(16,29,25,36)(17,32,33,30)(18,35,37,27)(21,31,38,44)(22,41,42,28)(23,43,34,40);(13,16,19,25)(14,21,24,38)(15,17,26,33)(18,32,37,30)(20,22,39,42)(23,41,34,28)(27,36,35,29)(31,40,44,43);(2,4)(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,25)(16,26)(18,36)(19,33)(20,38)(21,39)(23,44)(24,42)(27,30)(28,40)(29,37)(31,34)(32,35)(41,43);(6,9)(10,12)(15,26)(17,33)(20,39)(22,42)(27,35)(29,36)(31,44)(40,43)

Polycyclic presentation:

2^9|a^2=i,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,e^b=ei,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, 7423948);