#398914

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

959 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=f,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 306110);