#288914

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

654 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)(7,13)(8,17)(9,19)(10,12)(11,16)(14,18)(15,20)(21,22,25,31,27,30,26,32)(23,28,33,29,24,35,34,36);(1,2)(3,4)(5,7,9,16,10,15,18,8)(6,11,14,13,12,17,19,20)(21,23,25,33,27,24,26,34)(22,28,31,29,30,35,32,36);(2,4)(6,12)(7,15)(8,16)(14,19)(21,24,27,23)(22,29,30,36)(25,33,26,34)(28,32,35,31);(5,8,18,15,10,16,9,7)(6,13,19,11,12,20,14,17)(22,30)(28,35)(29,36)(31,32)

Polycyclic presentation:

2^9|a^2=fh,b^2=efh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,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, 196110);