#186856

order	512
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94052);