#186324

order	512
nilpotency class	3
isoclinism	isoclinic class of #186155
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

2053 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,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	#72239884
maximal normal subgroup of	#72239884
maximal quotient of	#72239900

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93520);