#278682

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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 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)(5,6,10,15,11,16,9,14)(7,12,17,13,8,18,19,20)(21,22,24,29,26,30,25,28)(23,27,33,35,32,34,31,36);(1,2,4,3)(5,7)(6,12)(8,11)(9,19)(10,17)(13,15)(14,20)(16,18)(21,23)(22,27)(24,33)(25,31)(26,32)(28,36)(29,35)(30,34);(5,8,11,7)(6,13,16,20)(9,19,10,17)(12,14,18,15)(22,28)(23,31)(24,25)(27,34)(29,30)(32,33);(1,3,4,2)(5,9,11,10)(6,14,16,15)(7,17,8,19)(12,13,18,20)(23,32)(27,34)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=hi,d^2=ehi,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, 185878);