#186249

order	512
nilpotency class	3
isoclinism	isoclinic class of #186176
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	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2091 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,26,23,20,21,25)(19,27,30,24,31,32,29,28);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19,23,31)(18,24,20,28)(21,29,22,30)(25,27,26,32);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,23)(19,28)(21,25)(22,26)(24,31)(27,29)(30,32);(2,7)(6,14)(8,15)(9,16)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,27,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239236
maximal normal subgroup of	#72239236
maximal quotient of	#72239258

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93445);