#499970

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

951 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)(5,6)(7,11)(8,10)(9,12)(13,14,20,24)(15,17,22,19)(16,27,31,42)(18,33,36,26)(21,38,40,30)(23,32,41,35)(25,34,39,44)(28,29,43,37);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,28)(18,23)(19,24)(20,22)(21,34)(25,30)(26,35)(27,29)(31,43)(32,33)(36,41)(37,42)(38,39)(40,44);(2,4)(13,16)(14,21)(15,25)(17,29)(18,34)(19,37)(20,31)(22,39)(23,28)(24,40)(26,42)(27,33)(30,35)(32,38)(36,44)(41,43);(6,10)(9,11)(13,17,20,19)(14,22,24,15)(16,29,31,37)(18,32,36,35)(21,39,40,25)(23,33,41,26)(27,43,42,28)(30,34,38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407166);