#242762

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

615 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=h,d^2=i,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, 149958);