#281753

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

750 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)(5,6,9,14,11,16,10,15)(7,12,17,13,8,19,18,20)(21,22,26,28,25,30,24,29)(23,27,33,36,31,35,32,34);(2,4)(5,7,9,17,11,8,10,18)(6,12,14,13,16,19,15,20)(21,23,26,33,25,31,24,32)(22,27,28,36,30,35,29,34);(5,8,11,7)(6,13,16,20)(9,17,10,18)(12,15,19,14)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(2,4)(21,24,25,26)(22,29,30,28)(23,32,31,33)(27,34,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188949);