#238599

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
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

957 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)(5,6,10,14)(7,11,17,20)(8,12,9,13)(15,18,16,19)(21,22)(23,27)(24,29)(25,28)(26,30)(31,34)(32,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,15)(10,17)(12,19)(13,18)(14,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,35)(29,34)(30,36);(1,3,4,2)(6,12)(7,15)(8,9)(13,14)(16,17)(18,19)(22,28)(24,25)(27,34)(29,30)(31,33)(35,36);(5,8,10,9)(6,13,14,12)(7,16,17,15)(11,18,20,19)(21,24,26,25)(22,28,30,29)(23,31,32,33)(27,35,36,34)

Polycyclic presentation:

2^9|a^2=hi,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145795);