#6445712

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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

2918 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)(15,24,27,40)(16,21,28,20)(17,30,31,22)(19,32,35,26)(25,39,41,38)(29,36,43,37)(33,42,34,44);(1,2)(3,4)(13,15,17,26,18,27,31,32)(14,19,22,24,23,35,30,40)(16,25,29,42,28,41,43,44)(20,33,36,38,21,34,37,39);(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)(24,38)(26,42)(27,41)(30,37)(31,43)(32,44)(34,35)(39,40);(2,4)(13,16)(14,21)(15,25)(17,29)(18,28)(19,34)(20,23)(22,37)(24,39)(26,42)(27,41)(30,36)(31,43)(32,44)(33,35)(38,40);(16,28)(20,21)(25,41)(29,43)(33,34)(36,37)(38,39)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6352908);