#225005

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

749 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)(9,10,12,14,11,13,15,16)(18,23)(19,21)(24,31)(25,30)(26,28)(27,32);(1,3,4,7)(2,5,6,8)(10,13)(14,16)(17,18,22,25)(19,23,21,30)(20,27,29,31)(24,28,32,26);(2,6)(5,8)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dei,e^b=eh,d^c=dh,e^d=eh

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, 132201);