#238587

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

933 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)(7,14)(8,18)(9,12)(10,13)(11,16)(15,20)(17,19)(21,22,26,31)(23,27,33,36)(24,30,25,29)(28,34,35,32);(1,2)(3,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,15)(16,20)(17,19)(21,23)(22,27)(24,32)(25,34)(26,33)(28,30)(29,35)(31,36);(1,3,4,2)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,20)(14,18)(21,23)(22,28)(24,32)(25,34)(26,33)(27,29)(30,36)(31,35);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,20,14)(21,24,26,25)(22,29,31,30)(23,32,33,34)(27,35,36,28)

Polycyclic presentation:

2^9|a^2=i,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,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, 145783);