#380107

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,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, 287303);