#235523

order	512
nilpotency class	3
isoclinism	isoclinic class of #235516
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1151 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142719);