#241058

order	512
nilpotency class	3
isoclinism	isoclinic class of #10560
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	#60949	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

942 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)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,30)(28,31)(29,32);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(6,12)(13,16)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)

Polycyclic presentation:

2^9|c^2=e,d^2=hi,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^c=gh

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, 148254);