#288846

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
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

673 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)(7,15)(8,11)(9,18)(10,12)(13,16)(14,19)(17,20)(21,22,25,29,24,28,31,27)(23,30,34,36,33,26,32,35);(1,2)(3,4)(5,7,9,8,10,16,19,17)(6,11,14,15,12,20,18,13)(21,23,31,32,24,33,25,34)(22,26,27,36,28,30,29,35);(2,4)(6,12)(7,16)(8,17)(14,18)(22,27)(23,32)(25,31)(26,30)(28,29)(33,34);(5,8,9,16,10,17,19,7)(6,13,14,11,12,15,18,20)(22,28)(26,30)(27,29)(35,36)

Polycyclic presentation:

2^9|a^2=fh,b^2=efh,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=dh,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196042);