#497992

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

556 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405188);