#190035

order	512
nilpotency class	3
isoclinism	isoclinic class of #190034
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

969 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97231);