#7614434

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1801 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)(7,11)(8,12)(9,10)(13,14,19,23)(15,25,28,21)(16,18,24,20)(17,31,35,33)(22,34,40,36)(26,39,43,44)(27,30,29,38)(32,37,42,41);(2,4)(6,10)(11,12)(13,15,19,28)(14,21,23,25)(16,30,24,38)(17,32,35,42)(18,27,20,29)(22,39,40,44)(26,36,43,34)(31,41,33,37);(5,7,9,8)(6,11,10,12)(13,16,18,23,19,24,20,14)(15,25,29,30,28,21,27,38)(17,33,36,22,35,31,34,40)(26,37,42,44,43,41,32,39);(1,2,3,4)(13,17,18,36,19,35,20,34)(14,22,16,31,23,40,24,33)(15,26,29,42,28,43,27,32)(21,37,38,44,25,41,30,39);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,16,23,24)(15,27,28,29)(17,34,35,36)(21,30,25,38)(22,33,40,31)(26,42,43,32)(37,44,41,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=hi,e^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,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521630);