#281622

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188818);