#238560

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 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)(7,8)(9,13)(10,14)(11,15)(12,19)(16,18)(17,20)(21,22,26,30)(23,27,32,36)(24,29,25,33)(28,34,35,31);(1,2)(3,4)(5,7)(6,8)(9,16)(10,12)(11,17)(13,18)(14,19)(15,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,29)(30,36)(33,35);(1,3,4,2)(5,8)(6,12)(7,13)(9,18)(10,19)(11,20)(14,17)(15,16)(21,23)(22,28)(24,31)(25,34)(26,32)(27,33)(29,36)(30,35);(5,9,11,10)(6,13,15,14)(7,16,17,12)(8,18,20,19)(21,24,26,25)(22,29,30,33)(23,31,32,34)(27,28,36,35)

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,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145756);