#399273

order	512
nilpotency class	3
isoclinism	isoclinic class of #395148
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

973 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,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, 306469);