#242994

order	512
nilpotency class	3
isoclinism	isoclinic class of #242992
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

644 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)(4,8)(5,7)(9,13)(10,15)(11,14)(12,16)(17,18,22,27,23,28,21,26)(19,29,31,24,33,40,32,41)(20,34,36,42,37,43,38,25)(30,45,47,48,35,39,46,44);(1,3)(2,6)(4,9)(5,11)(7,14)(8,13)(10,12)(15,16)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,39)(26,40)(27,29)(28,41)(30,37)(34,48)(36,47)(38,46)(42,45)(43,44);(3,9)(4,12)(6,13)(8,16)(10,11)(14,15)(17,20,21,36,23,37,22,38)(18,25,26,34,28,42,27,43)(19,30,32,47,33,35,31,46)(24,39,29,44,41,45,40,48);(2,7)(6,14)(8,16)(13,15)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,34,42,43)(30,46,35,47)(39,48,45,44)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,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, 150190);