#6491542

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2913 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)(13,14,18,23,20,25,19,24)(15,26,28,36,16,30,29,21)(17,22,33,39,32,38,34,40)(27,41,42,37,31,44,43,35);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,16)(14,21,25,36)(17,31,32,27)(18,29,19,28)(22,37,38,35)(23,30,24,26)(33,42,34,43)(39,41,40,44);(6,9)(10,12)(13,16,20,15)(14,21,25,36)(17,31,32,27)(18,28,19,29)(22,35,38,37)(23,30,24,26)(33,42,34,43)(39,44,40,41);(6,9)(10,12)(13,17)(14,22)(15,27)(16,31)(18,33)(19,34)(20,32)(21,35)(23,39)(24,40)(25,38)(26,41)(28,42)(29,43)(30,44)(36,37);(5,7,8,11)(6,10,9,12)(17,32)(22,38)(27,31)(33,34)(35,37)(39,40)(41,44)(42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dhi,f^b=fi,g^b=gi,f^c=fi,g^c=gi,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, 6398738);