#509871

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

534 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417067);