#223397

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

899 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=de,f^c=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130593);