#399224

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1376 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)(13,14,17,21,19,24,18,23)(15,25,26,36,29,35,28,20)(16,30,31,22,33,37,32,38)(27,41,42,39,43,34,40,44);(2,4)(5,7,9,8)(6,10,11,12)(13,15,19,29)(14,20,24,36)(16,27,33,43)(17,28,18,26)(21,35,23,25)(22,34,38,41)(30,44,37,39)(31,40,32,42);(2,4)(5,8,9,7)(6,10,11,12)(14,21)(15,26)(17,18)(22,30)(23,24)(25,35)(27,40)(28,29)(31,32)(37,38)(39,44)(42,43);(13,16)(14,22)(15,27)(17,32)(18,31)(19,33)(20,34)(21,30)(23,37)(24,38)(25,39)(26,40)(28,42)(29,43)(35,44)(36,41)

Polycyclic presentation:

2^9|a^2=e,b^2=efi,c^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 306420);