#345834

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
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

499 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)(20,35)(21,27)(22,26)(23,28)(24,32)(25,37)(30,45)(31,34)(33,41)(36,46)(38,43)(39,44)(40,48)(42,47);(3,9)(4,11)(6,13)(7,15)(10,12)(14,16)(17,19,22,33,23,34,21,32)(18,24,27,31,28,41,26,29)(20,36,37,30,39,47,38,48)(25,42,35,45,43,46,44,40);(1,3,5,10)(2,6,8,14)(4,12,11,9)(7,16,15,13)(17,20)(18,25)(19,30)(21,37)(22,38)(23,39)(24,40)(26,35)(27,44)(28,43)(29,42)(31,46)(32,47)(33,36)(34,48)(41,45);(17,18)(19,31)(20,25)(21,27)(22,26)(23,28)(24,33)(29,34)(30,46)(32,41)(35,38)(36,40)(37,44)(39,43)(42,48)(45,47)

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=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253030);