#497012

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

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

618 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:

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

Polycyclic presentation:

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

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, 404208);