#298291

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fhi,d^2=e,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,g^c=gh

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, 205487);