#6491139

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3171 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6398335);