#508011

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
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

999 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)(13,14,17,22)(15,25,16,20)(18,32,30,36)(19,35,31,24)(21,26,39,28)(23,42,41,34)(27,38,29,43)(33,37,44,40);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,17)(18,26)(19,27)(21,32)(22,25)(23,37)(24,38)(28,30)(29,31)(33,42)(34,44)(35,43)(36,39)(40,41);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16,17,15)(14,21,22,39)(18,33,30,44)(19,27,31,29)(20,36,25,32)(23,38,41,43)(24,37,35,40)(26,34,28,42);(7,11)(10,12)(15,16)(20,25)(21,39)(26,28)(27,29)(33,44)(37,40)(38,43)

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=cg,d^b=di,d^c=de,f^c=fh,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, 415207);