#363486

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 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)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,24,20,26,18,25)(15,23,17,35,29,37,28,21)(16,30,32,22,34,39,33,40)(27,41,42,38,43,36,31,44);(5,7,8,12)(6,9,10,11)(13,15,18,28,20,29,19,17)(14,21,24,23,26,35,25,37)(16,27,32,42,34,43,33,31)(22,36,30,38,40,41,39,44);(2,4)(7,12)(9,11)(13,16)(14,22)(15,27)(17,31)(18,32)(19,33)(20,34)(21,36)(23,38)(24,30)(25,39)(26,40)(28,42)(29,43)(35,41)(37,44);(6,10)(9,11)(13,17,19,29,20,28,18,15)(14,23,25,21,26,37,24,35)(16,31,33,43,34,42,32,27)(22,38,39,36,40,44,30,41)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270682);