#238571

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

887 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)(7,11)(8,12)(9,13)(10,14)(15,19)(16,18)(17,20)(21,22)(23,27)(24,28)(25,33)(26,29)(30,31)(32,35)(34,36);(1,2)(3,4)(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,18,13,19)(21,23)(22,27)(24,31)(25,34)(26,32)(28,30)(29,35)(33,36);(1,3,4,2)(6,12)(7,15)(8,9)(13,14)(16,17)(18,19)(21,22)(23,30)(24,28)(25,33)(26,29)(27,34)(31,35)(32,36);(5,8,10,9)(6,12,14,13)(7,16,17,15)(11,18,20,19)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,30,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145767);