#6444968

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352164);