#288564

order	512
nilpotency class	3
isoclinism	isoclinic class of #264555
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

684 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=ef,c^2=i,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=gi,d^c=dhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195760);