#512092

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,d^b=di,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=gh,h^d=hi

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, 419288);