#268798

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3247	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175994);