#186873

order	512
nilpotency class	3
isoclinism	isoclinic class of #186808
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62781	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2213 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306301, #72306303, #72306337
maximal normal subgroup of	#72306301, #72306303, #72306337

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94069);