#288009

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

653 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)(5,6,10,14,9,13,16,19)(7,15,17,12,8,11,18,20)(21,22,27,33,26,24,25,29)(23,28,34,36,31,32,30,35);(1,2,4,3)(5,7,16,18,9,8,10,17)(6,11,19,12,13,15,14,20)(21,23,25,30,26,31,27,34)(22,28,29,35,24,32,33,36);(5,8,9,7)(6,12,13,20)(10,17,16,18)(11,14,15,19)(21,24,26,22)(23,28,31,32)(25,29,27,33)(30,36,34,35);(1,3,4,2)(22,24)(28,32)(29,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195205);