#312212

order	512
nilpotency class	3
isoclinism	isoclinic class of #244927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

547 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=h,c^2=h,d^2=efh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219408);