#261030

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

710 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)(4,5)(6,8)(9,10,13,18)(11,19,22,15)(12,16,14,17)(20,24,21,23)(25,26,29,33)(27,35,28,31)(30,39,38,34)(32,36,40,37);(2,5)(6,7)(9,11,13,22)(10,15,18,19)(12,21,14,20)(16,24,17,23)(25,27,29,28)(26,31,33,35)(30,37,38,36)(32,39,40,34);(10,16)(11,20)(12,14)(17,18)(21,22)(23,24)(25,28,29,27)(26,32,33,40)(30,36,38,37)(31,34,35,39);(1,3,4,8)(2,6,5,7)(9,12,13,14)(10,17,18,16)(11,21,22,20)(15,23,19,24)(26,33)(31,35)(32,40)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=ei,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

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, 168226);