#208512

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

841 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)(10,13)(11,15)(14,16)(17,18,20,25,21,26,29,24)(19,23,28,22,27,31,32,30);(9,10,11,13,12,14,15,16)(17,19,21,27)(18,22)(20,28,29,32)(23,24)(25,31)(26,30);(2,5)(6,8)(10,14)(13,16)(17,19)(18,23)(20,28)(21,27)(22,25)(24,30)(26,31)(29,32);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(19,27)(22,30)(23,31)(28,32)

Polycyclic presentation:

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

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, 115708);