#7156269

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2008 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)(15,24)(16,22)(17,23)(18,20)(19,32)(21,26)(25,39)(27,36)(28,34)(29,37)(30,33)(31,38)(35,42)(40,44)(41,43);(2,3)(6,9)(7,11)(14,20)(16,28)(21,33)(23,32)(24,36)(25,40)(29,31)(41,42);(5,6)(7,10)(8,9)(11,12)(13,15,17,30,18,27,19,26)(14,21,32,36,20,33,23,24)(16,25,31,41,28,40,29,42)(22,35,37,44,34,43,38,39);(1,3,4,2)(6,9)(10,12)(13,16,18,28)(14,22,20,34)(15,25,27,40)(17,31,19,29)(21,35,33,43)(23,38,32,37)(24,39,36,44)(26,42,30,41);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,20,32)(15,26,27,30)(16,29,28,31)(21,36,33,24)(22,37,34,38)(25,41,40,42)(35,39,43,44)

Polycyclic presentation:

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

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, 7063465);