#187425

order	512
nilpotency class	3
isoclinism	isoclinic class of #187419
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	#62834	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2389 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)(3,11,14,7)(4,15)(5,9)(6,10)(8,13,16,12)(18,23,25,28)(24,31,32,30);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,11)(15,16)(17,18)(19,24)(20,23)(21,28)(22,25)(26,30)(27,32)(29,31);(1,3)(2,8)(4,13)(5,12)(6,14)(7,9)(10,16)(11,15)(17,19)(18,24)(20,26)(21,29)(22,27)(23,30)(25,32)(28,31);(1,4,6,5)(2,9,10,15)(3,12,14,13)(7,8,11,16)(17,20,22,21)(18,23,25,28)(19,26,27,29)(24,30,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238922, C₂×#187425
maximal normal subgroup of	#72238922, C₂×#187425
maximal quotient of	C₂×#187425, #72238963, #72238967

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94621);