#190432

order	512
nilpotency class	3
isoclinism	isoclinic class of #190432
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	C₂⁶	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1380 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,g^2=i,b^a=be,c^a=cf,e^a=eh,g^b=gh,d^c=di,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, 97628);