#399229

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

Polycyclic presentation:

2^9|a^2=ei,b^2=efi,c^2=efi,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, 306425);