#242974

order	512
nilpotency class	3
isoclinism	isoclinic class of #242971
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

596 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,22,27,23,28,21,26)(19,29,32,24,33,40,31,41)(20,34,36,42,37,43,38,25)(30,45,46,48,35,39,47,44);(1,3)(2,6)(4,10)(5,9)(7,14)(8,16)(11,12)(13,15)(17,19,23,33)(18,24,28,41)(20,35,37,30)(21,32,22,31)(25,39,42,45)(26,40,27,29)(34,48,43,44)(36,46,38,47);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20,21,36,23,37,22,38)(18,25,26,34,28,42,27,43)(19,30,31,46,33,35,32,47)(24,39,29,44,41,45,40,48);(3,10)(6,14)(9,11)(13,16)

Polycyclic presentation:

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

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, 150170);