#345874

order	512
nilpotency class	3
isoclinism	isoclinic class of #345848
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

502 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

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, 253070);