#235184

order	512
nilpotency class	3
isoclinism	isoclinic class of #235159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1284 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142380);