#247998

order	512
nilpotency class	3
isoclinism	isoclinic class of #247975
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 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:

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

Polycyclic presentation:

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

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, 155194);