#7518001

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

1815 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)(13,14,20,26)(15,21,31,40)(16,32,34,42)(17,35,37,44)(18,24,19,25)(22,28,43,33)(23,27,41,36)(29,38,30,39);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,31)(14,21,26,40)(16,17,34,37)(18,29,19,30)(22,41,43,23)(24,38,25,39)(27,28,36,33)(32,35,42,44);(13,16,20,34)(14,22,26,43)(15,27,31,36)(17,30,37,29)(18,28,19,33)(21,35,40,44)(23,39,41,38)(24,32,25,42);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,28)(16,29)(18,27)(19,36)(20,37)(21,32)(22,38)(24,35)(25,44)(26,41)(30,34)(31,33)(39,43)(40,42);(6,11)(10,12)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,33,34,28)(17,27,37,36)(21,38,40,39)(22,42,43,32)(23,35,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,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,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, 7425197);