#395357

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

1400 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)(13,14,17,22)(15,25,28,20)(16,30,31,40)(18,24,19,23)(21,34,38,32)(26,42,43,36)(27,41,44,35)(29,37,33,39);(5,7,9,8)(6,10,11,12)(13,15,17,28)(14,20,22,25)(16,27,31,44)(18,33,19,29)(21,36,38,42)(23,39,24,37)(26,34,43,32)(30,35,40,41);(2,4)(13,16)(14,21)(15,26)(17,31)(18,34)(19,32)(20,35)(22,38)(23,40)(24,30)(25,41)(27,33)(28,43)(29,44)(36,39)(37,42);(5,8,9,7)(6,10,11,12)(13,16)(14,21)(15,27)(17,31)(18,32)(19,34)(20,36)(22,38)(23,30)(24,40)(25,42)(26,33)(28,44)(29,43)(35,39)(37,41)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,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, 302553);