#508048

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1218 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)(15,19)(16,20)(17,31)(18,21)(22,28)(23,29)(24,34)(25,38)(26,39)(27,33)(30,36)(32,35)(37,42)(40,44)(41,43);(6,9)(10,12)(13,15)(14,19)(16,24)(17,32)(18,27)(20,34)(21,33)(22,38)(23,39)(25,28)(26,29)(30,41)(31,35)(36,43)(37,44)(40,42);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,24)(16,28)(17,29)(21,36)(22,23)(25,26)(27,41)(30,42)(31,37)(32,40)(34,38)(35,39)(43,44);(7,11)(10,12)(14,21)(19,33)(20,36)(22,37)(23,31)(34,43)(35,39)(38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415244);