#7145950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45387	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2193 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7053146);