#381926

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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)(6,9)(10,12)(13,14,19,24,17,22,18,23)(15,25,29,36,27,20,28,37)(16,30,34,21,32,40,33,39)(26,38,31,41,42,44,43,35);(5,6,8,9)(7,10,11,12)(13,15,17,27)(14,20,22,25)(16,31,32,43)(18,28,19,29)(21,38,39,44)(23,36,24,37)(26,34,42,33)(30,35,40,41);(7,11)(10,12)(13,16,17,32)(14,21,22,39)(15,26,27,42)(18,34,19,33)(20,35,25,41)(23,40,24,30)(28,31,29,43)(36,38,37,44);(1,3,4,2)(5,7,8,11)(6,10,9,12)(16,32)(21,39)(26,42)(30,40)(31,43)(33,34)(35,41)(38,44)

Polycyclic presentation:

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

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, 289122);