#496589

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

840 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fh,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403785);