#190049

order	512
nilpotency class	3
isoclinism	isoclinic class of #190047
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1018 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,26,27,22)(20,23)(21,24)(25,29)(28,31,32,30);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19,21,27)(18,22,24,26)(20,28,29,32)(23,30,25,31);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(18,23,24,25)(22,30,26,31);(19,27)(22,26)(28,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97245);