#6445395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3042 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6352591);