#6617443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25727	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3013 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)(13,14,19,33,18,22,17,23)(15,24,31,20,26,37,27,35)(16,28,32,21,30,38,29,36)(25,34,42,39,40,43,41,44);(2,4)(6,10)(8,11)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,34)(22,35)(23,37)(24,33)(28,39)(29,42)(30,40)(32,41)(36,43)(38,44);(5,6)(7,8)(9,10)(11,12)(13,16,18,30)(14,21,22,36)(15,25,26,40)(17,32,19,29)(20,34,35,43)(23,38,33,28)(24,39,37,44)(27,42,31,41);(5,7,9,12)(6,11,10,8)(15,26)(20,35)(24,37)(25,40)(27,31)(34,43)(39,44)(41,42);(5,8,9,11)(6,7,10,12)(14,22)(20,35)(21,36)(23,33)(24,37)(28,38)(34,43)(39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524639);