#7145789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2155 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=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, 7052985);