#187950

order	512
nilpotency class	3
isoclinism	isoclinic class of #187949
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	#62818	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2622 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234608, #72234610, #72234611, …
maximal normal subgroup of	#72234608, #72234610, #72234611, …
maximal quotient of	#72313592

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95146);