#7514808

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	#35687	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2655 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)(8,10)(9,12)(13,14,18,23)(15,25,26,20)(16,28,30,43)(17,32,33,42)(19,35,34,24)(21,37,40,31)(22,38,39,29)(27,41,36,44);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,29)(17,31)(18,26)(19,36)(21,42)(22,43)(23,25)(24,44)(27,34)(28,39)(30,38)(32,40)(33,37)(35,41);(13,16,18,30)(14,21,23,40)(15,17,26,33)(19,37,34,31)(20,39,25,22)(24,43,35,28)(27,38,36,29)(32,44,42,41);(2,4)(13,17)(14,22)(15,16)(18,33)(19,38)(20,40)(21,25)(23,39)(24,42)(26,30)(27,37)(28,41)(29,34)(31,36)(32,35)(43,44);(6,10)(9,11)(14,23)(15,26)(17,33)(21,40)(24,35)(27,36)(28,43)(29,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^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,g^c=gi,h^c=hi,e^d=ei,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, 7422004);