#7545812

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7453008);