#185772

order	512
nilpotency class	3
isoclinism	isoclinic class of #185475
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2342 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:

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

Polycyclic presentation:

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

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, 92968);