#310408

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

680 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)(3,6)(4,5)(7,8)(9,10,12,14,13,15,11,16)(17,18,21,22)(19,23,26,30)(20,27,29,24)(25,28,32,31);(9,10)(11,14)(12,16)(13,15)(17,19,21,26)(18,23,22,30)(20,28,29,31)(24,25,27,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,21,29)(18,24,22,27)(19,25,26,32)(23,31,30,28);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,16)(13,15)(20,29)(24,27)(25,32)(28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217604);