#278721

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 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)(5,6,11,14,10,15,9,16)(7,17,18,20,8,12,19,13)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36);(1,2,4,3)(5,7,10,8)(6,12,15,17)(9,19,11,18)(13,16,20,14)(21,23)(22,26)(24,31)(25,30)(27,29)(28,34)(32,33)(35,36);(5,8,10,7)(6,13,15,20)(9,19,11,18)(12,16,17,14)(23,30)(25,33)(26,34)(28,36)(29,35)(31,32);(1,3,4,2)(5,9,10,11)(6,14,15,16)(7,18,8,19)(12,20,17,13)(22,27)(26,29)(28,36)(34,35)

Polycyclic presentation:

2^9|a^2=fh,b^2=ei,c^2=h,d^2=ei,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, 185917);