#298319

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

906 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)(5,6,9,12)(7,14,16,11)(8,18,13,10)(15,19,20,17)(21,22,25,28)(23,26,30,35)(24,32,33,27)(29,36,31,34);(1,2,4,3)(5,7,9,16)(6,11,12,14)(8,15,13,20)(10,19,18,17)(21,23)(22,26)(24,29)(25,30)(27,34)(28,35)(31,33)(32,36);(5,8,10,6,9,13,18,12)(7,15,17,14,16,20,19,11)(23,29,30,31)(26,34,35,36);(1,3,4,2)(6,12)(8,13)(11,14)(15,20)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,d^2=eh,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=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205515);