#345125

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ehi,c^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 252321);