#396480

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1244 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 303676);