#242750

order	512
nilpotency class	3
isoclinism	isoclinic class of #242745
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 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:

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

Polycyclic presentation:

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

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, 149946);