#289945

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×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,c^2=hi,d^2=ei,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, 197141);