#242757

order	512
nilpotency class	3
isoclinism	isoclinic class of #242755
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

668 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,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,d^c=dh,e^c=eh,f^c=fi

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, 149953);