#279981

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
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	#24116	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

787 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 187177);