#7516624

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 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)(16,27,32,28)(17,33,34,38)(18,24,19,23)(21,31,25,30)(22,36,26,35)(29,37,42,39)(40,44,41,43);(1,2)(3,4)(6,9)(10,12)(13,15,20,14)(16,28,32,27)(17,33,34,38)(18,23,19,24)(21,30,25,31)(22,36,26,35)(29,37,42,39)(40,44,41,43);(13,16,20,32)(14,21,15,25)(17,29,34,42)(18,31,19,30)(22,37,26,39)(23,28,24,27)(33,40,38,41)(35,44,36,43);(2,4)(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,26)(16,29)(18,36)(19,35)(20,34)(21,37)(23,38)(24,33)(25,39)(27,40)(28,41)(30,44)(31,43)(32,42);(6,9)(10,12)(17,34)(22,26)(29,42)(33,38)(35,36)(37,39)(40,41)(43,44)

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,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, 7423820);