#7515667

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

1790 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)(7,12)(8,10)(9,11)(13,14,18,20)(15,25,28,36)(16,29,32,38)(17,24,19,23)(21,33,34,26)(22,31,35,30)(27,37,42,43)(39,40,44,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,20)(15,26)(16,30)(17,19)(21,36)(22,38)(25,34)(28,33)(29,35)(31,32)(37,43)(40,41);(5,8,9,7)(6,10,11,12)(13,15)(14,21)(16,27)(17,26)(18,28)(19,33)(20,34)(22,37)(23,25)(24,36)(29,39)(30,40)(31,41)(32,42)(35,43)(38,44);(2,4)(13,16,18,32)(14,22,20,35)(15,27,28,42)(17,30,19,31)(21,37,34,43)(23,29,24,38)(25,39,36,44)(26,40,33,41);(6,11)(10,12)(13,17,18,19)(14,23,20,24)(15,26,28,33)(16,31,32,30)(21,25,34,36)(22,38,35,29)(27,41,42,40)(37,44,43,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=i,e^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,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, 7422863);