#187898

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2132 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240112, #72240115, #72240121
maximal normal subgroup of	#72240112, #72240115, #72240121

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95094);