#298365

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
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	#45999	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=eh,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=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205561);