#401448

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

Aliases

No aliases for this group.

Subgroups

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

1075 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 308644);