#289910

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

661 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=efh,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 197106);