#7515274

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

1931 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)(16,27,30,28)(17,33,37,40)(18,24,19,23)(21,32,25,31)(22,36,26,35)(29,38,34,39)(41,44,42,43);(1,2)(3,4)(6,9)(10,12)(13,15,20,14)(16,28,30,27)(17,33,37,40)(18,23,19,24)(21,31,25,32)(22,36,26,35)(29,38,34,39)(41,44,42,43);(5,7,8,11)(6,10,9,12)(13,16,20,30)(14,21,15,25)(17,34,37,29)(18,32,19,31)(22,39,26,38)(23,28,24,27)(33,42,40,41)(35,43,36,44);(2,4)(13,17,20,37)(14,22,15,26)(16,29,30,34)(18,36,19,35)(21,38,25,39)(23,40,24,33)(27,41,28,42)(31,44,32,43);(6,9)(10,12)(16,30)(21,25)(27,28)(29,34)(31,32)(38,39)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,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, 7422470);