#6445653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2134 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)(7,11)(8,9)(10,12)(13,14,19,32,18,23,17,24)(15,25,31,20,28,39,27,36)(16,22,33,37,29,21,30,38)(26,41,44,35,42,40,43,34);(1,2)(3,4)(6,9)(10,11)(13,15,18,28)(14,20,23,36)(16,26,29,42)(17,31,19,27)(21,34,22,35)(24,39,32,25)(30,44,33,43)(37,41,38,40);(2,4)(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,26)(17,30)(18,29)(19,33)(20,34)(22,23)(24,37)(25,40)(27,43)(28,42)(31,44)(32,38)(35,36)(39,41);(2,4)(13,16)(14,22)(15,26)(17,30)(18,29)(19,33)(20,35)(21,23)(24,38)(25,41)(27,43)(28,42)(31,44)(32,37)(34,36)(39,40);(14,23)(16,29)(20,36)(24,32)(25,39)(26,42)(30,33)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352849);