#496811

order	512
nilpotency class	4
isoclinism	isoclinic class of #33752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 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)(17,29)(18,33)(19,23)(20,26)(22,31)(25,40)(27,41)(28,39)(30,35)(32,43)(34,38)(36,44)(37,42);(1,3,4,2)(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(13,15)(14,20)(16,25)(17,30)(18,34)(19,27)(21,35)(22,38)(23,37)(24,29)(26,31)(28,42)(32,44)(33,36)(39,40)(41,43);(6,10)(9,11)(13,16,19,18)(14,21,23,33)(15,25,27,34)(17,28,32,31)(20,35,37,36)(22,29,39,43)(24,40,41,38)(26,30,42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,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,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, 404007);