#438617

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

823 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=ef,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345813);