#223007

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
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

862 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)(10,14)(11,12)(15,16)(17,18,21,24,20,22,27,29)(19,23,26,30,25,28,31,32);(1,2)(3,6)(4,5)(7,8)(10,15)(14,16)(18,22)(19,25)(24,29)(26,31);(1,3,4,7)(2,6,5,8)(9,10,13,15)(11,14,12,16)(17,19,20,25)(18,23,22,28)(21,26,27,31)(24,30,29,32);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,22)(19,25)(24,29)(26,31)

Polycyclic presentation:

2^9|a^2=g,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,c^b=chi,d^b=dei,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130203);