#224628

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=e,d^2=ehi,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131824);