#400172

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1466 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)(5,6,8,9)(7,10,11,12)(13,14,18,23,20,24,19,36)(15,25,28,40,29,39,35,21)(16,30,32,41,17,33,34,22)(26,37,31,38,44,43,27,42);(2,4)(6,9)(10,12)(13,15,20,29)(14,21,24,40)(16,31,17,27)(18,35,19,28)(22,38,41,42)(23,39,36,25)(26,34,44,32)(30,37,33,43);(2,4)(13,16,20,17)(14,22,24,41)(15,26,29,44)(18,34,19,32)(21,37,40,43)(23,33,36,30)(25,42,39,38)(27,35,31,28);(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,22,24,41)(15,27,29,31)(18,34,19,32)(21,38,40,42)(23,30,36,33)(25,43,39,37)(26,35,44,28)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,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, 307368);