#235577

order	512
nilpotency class	3
isoclinism	isoclinic class of #235570
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	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,14)(10,16)(13,15)(17,18,23,28)(19,29,32,40)(20,25,35,42)(21,27,22,26)(24,33,39,31)(30,45,46,41)(34,48,47,38)(36,44,37,43);(6,15)(9,16)(17,19,23,32)(18,24,28,39)(20,34,35,47)(21,33,22,31)(25,41,42,45)(26,40,27,29)(30,37,46,36)(38,44,48,43);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,12)(11,14)(17,20,23,35)(18,25,28,42)(19,30,32,46)(21,36,22,37)(24,38,39,48)(26,43,27,44)(29,41,40,45)(31,34,33,47);(3,10)(6,15)(9,16)(11,13)(20,35)(25,42)(30,46)(34,47)(36,37)(38,48)(41,45)(43,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,d^c=dhi,e^c=ei,f^c=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, 142773);