#187920

order	512
nilpotency class	3
isoclinism	isoclinic class of #187878
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2113 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,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, #72240121, #72268187, …
maximal normal subgroup of	#72240112, #72240121, #72268187, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95116);