#508158

order	512
nilpotency class	4
isoclinism	isoclinic class of #507998
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
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^c=de,f^c=fh,h^c=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, 415354);