#298294

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	#45999	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1085 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)(7,15)(8,18)(9,12)(10,14)(11,16)(13,19)(17,20)(21,22)(23,29)(24,26)(25,33)(27,32)(28,31)(30,35)(34,36);(1,2,4,3)(5,7)(6,11)(8,10)(9,16)(12,15)(13,18)(14,20)(17,19)(22,26)(23,30)(25,31)(27,34);(5,8,10,16,9,17,19,7)(6,11,13,18,12,15,14,20)(21,23,31,36,24,30,25,32)(22,27,33,35,26,34,28,29);(1,3,4,2)(6,12)(11,15)(13,14)(18,20)(22,26)(27,34)(28,33)(29,35)

Polycyclic presentation:

2^9|b^2=eh,c^2=fi,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,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, 205490);