#240264

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 147460);