#187939

order	512
nilpotency class	3
isoclinism	isoclinic class of #187934
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#62819	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2390 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)(3,9,11,6)(4,7)(5,8)(10,13,16,15)(12,14)(22,28)(29,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,28)(27,30)(31,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(4,12)(7,14)(10,16)(13,15)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313569
maximal normal subgroup of	#72313569
maximal quotient of	#72313588

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95135);