#186202

order	512
nilpotency class	3
isoclinism	isoclinic class of #186137
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3520	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249679, #72249767
maximal normal subgroup of	#72249679, #72249767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93398);