#208491

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#2428	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

996 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)(9,10,14,15,13,11,12,16)(18,22,25,28)(19,26)(24,31,32,29)(27,30);(9,11,13,10)(12,16,14,15)(17,18,23,22,21,25,20,28)(19,24,30,29,26,32,27,31);(2,5)(6,8)(9,12,13,14)(10,15,11,16)(17,19,21,26)(18,24,25,32)(20,27,23,30)(22,29,28,31);(1,3,4,7)(2,6,5,8)(19,26)(24,32)(27,30)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=h,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, 115687);