#208485

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 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)(9,10,13,14,12,15,11,16)(18,22,24,27)(19,25)(23,30,31,29)(26,32);(10,14)(11,13)(15,16)(17,18,20,27,21,24,28,22)(19,23,26,30,25,31,32,29);(2,5)(6,8)(10,15)(14,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,31)(27,30)(28,32);(1,3,4,7)(2,6,5,8)(19,25)(23,31)(26,32)(29,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=dhi

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, 115681);