#506554

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

701 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)(7,11)(8,9)(10,12)(13,14,17,22,29,38,44,36,16,21,28,37,18,23,30,20)(15,24,26,39,41,34,42,35,25,19,31,33,27,40,43,32);(1,2,3,4)(6,9)(10,11)(13,15)(14,19)(16,25)(17,26)(18,27)(20,35)(21,24)(22,33)(23,34)(28,31)(29,41)(30,43)(32,36)(37,39)(38,40)(42,44);(2,4)(14,20)(15,25)(17,30)(18,29)(19,32)(21,36)(22,23)(24,35)(26,42)(28,44)(31,43)(33,40)(34,39)(37,38);(5,7,8,12)(6,10,9,11)(14,21)(19,24)(20,36)(22,37)(23,38)(32,35)(33,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413750);