#7545906

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 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)(15,25)(16,29)(17,20)(18,36)(19,22)(21,31)(23,28)(24,26)(27,42)(30,40)(32,37)(33,41)(34,39)(35,38)(43,44);(1,2)(3,4)(5,7,9,6)(8,10,12,11)(13,15,26,35,19,28,18,17)(14,20,36,23,22,38,24,25)(16,30,43,34,31,27,33,32)(21,39,44,40,29,37,41,42);(5,8,9,12)(6,10,7,11)(13,16,19,31)(14,21,22,29)(15,27,28,30)(17,34,35,32)(18,33,26,43)(20,37,38,39)(23,42,25,40)(24,41,36,44);(2,4)(6,7)(10,11)(14,22)(15,28)(16,31)(17,35)(24,36)(33,43)(37,39)(40,42);(13,17,18,28,19,35,26,15)(14,23,24,20,22,25,36,38)(16,32,33,27,31,34,43,30)(21,40,41,39,29,42,44,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=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, 7453102);