#289512

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 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)(5,6,9,14,11,16,10,15)(7,12,18,13,8,19,17,20)(21,22,26,28,24,29,25,30)(23,31,34,35,32,27,33,36);(1,2,4,3)(5,7,9,18,11,8,10,17)(6,12,14,13,16,19,15,20)(21,23,26,34,24,32,25,33)(22,27,28,36,29,31,30,35);(5,8,11,7)(6,13,16,20)(9,18,10,17)(12,15,19,14)(22,28)(23,32)(25,26)(27,35)(29,30)(31,36);(1,3,4,2)(5,9,11,10)(6,14,16,15)(7,17,8,18)(12,20,19,13)(22,29)(23,32)(28,30)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196708);