#325405

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 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)(3,7)(4,6)(5,8)(11,17)(12,20)(14,21)(16,22)(18,23)(19,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,38)(36,40)(37,39);(1,3,5,4)(2,6,8,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,23)(20,22)(21,24)(25,27,29,36,28,35,37,34)(26,30,32,38,31,33,39,40);(1,3,5,4)(2,7,8,6)(9,11,12,17,13,19,20,24)(10,14,16,21,15,18,22,23)(26,31)(27,34)(29,37)(30,38)(33,40)(35,36);(1,4,5,3)(2,7,8,6)(10,15)(14,18)(16,22)(21,23)(26,31)(30,33)(32,39)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 232601);