#281604

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

761 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)(21,22,25,34,26,24,27,29)(23,30,31,35,32,28,33,36);(2,4)(5,7,10,18,9,17,8,16)(6,11,12,20,13,14,19,15)(21,23,27,33,26,32,25,31)(22,28,29,35,24,30,34,36);(5,6)(7,15)(8,12)(9,13)(10,19)(11,16)(14,18)(17,20)(21,24,26,22)(23,28,32,30)(25,34,27,29)(31,35,33,36);(2,4)(5,8,9,10)(6,12,13,19)(7,16,17,18)(11,20,14,15)(21,25,26,27)(22,29,24,34)(23,31,32,33)(28,35,30,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=h,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, 188800);