#187946

order	512
nilpotency class	3
isoclinism	isoclinic class of #187944
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	#62773	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2589 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210515, #72213569, #72213643, …
maximal normal subgroup of	#72210515, #72213569, #72213643, …
maximal quotient of	#72313588, #72313592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95142);