#187933

order	512
nilpotency class	3
isoclinism	isoclinic class of #187928
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

2617 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)(4,15)(8,20,23,18)(9,17,21,29)(14,25)(16,26)(22,30,31,28);(1,2)(3,8)(4,7)(5,10)(6,9)(11,23)(12,21)(13,18)(14,22)(15,19)(16,17)(20,24)(25,31)(26,29)(27,28)(30,32);(4,16)(7,17)(13,27)(15,26)(18,28)(19,29)(20,30)(24,32);(1,3,12,25)(2,8,21,31)(4,13,26,32)(5,11,6,14)(7,18,29,30)(9,22,10,23)(15,24,16,27)(17,28,19,20)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210515, #72213613, #72213681, …
maximal normal subgroup of	#72210515, #72213613, #72213681, …
maximal quotient of	#72313588, #72313592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95129);