#236258

order	512
nilpotency class	3
isoclinism	isoclinic class of #10088
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1210 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)(3,8,12,15)(4,9,5,13)(7,14,16,11)(17,18,22,19)(20,27,28,32)(21,26,31,24)(23,30,25,29);(1,3,6,12)(2,7,10,16)(4,11,5,14)(8,9,15,13)(17,19,22,18)(20,25,28,23)(21,24,31,26)(27,30,32,29);(1,3)(2,8)(4,11)(5,14)(6,12)(7,9)(10,15)(13,16)(17,20)(18,23)(19,25)(21,30)(22,28)(24,32)(26,27)(29,31);(1,4,6,5)(2,9,10,13)(3,11,12,14)(7,15,16,8)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143454);