#190038

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1003 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18,21,25,22,26,20,24)(19,23,27,30,28,31,29,32);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,19)(18,23)(20,29)(21,27)(22,28)(24,32)(25,30)(26,31);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16);(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=i,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, 97234);