#346031

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

501 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)(3,8)(4,12)(5,7)(6,9)(10,16)(11,14)(13,15)(17,18,24,30)(19,31,34,25)(20,36,21,38)(22,29,23,28)(26,40,27,37)(32,43,48,45)(33,42,39,47)(35,44,41,46);(2,7)(3,10)(4,6)(8,15)(11,13)(14,16)(17,19,22,41,24,34,23,35)(18,25,28,46,30,31,29,44)(20,33,37,48,21,39,40,32)(26,43,36,47,27,45,38,42);(1,3)(2,8)(4,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20,24,21)(18,26,30,27)(19,32)(22,40,23,37)(25,42)(28,38,29,36)(31,47)(33,35)(34,48)(39,41)(43,44)(45,46);(1,4,5,6)(2,9,7,12)(3,10,11,13)(8,15,14,16)(17,21,24,20)(18,27,30,26)(19,33,34,39)(22,37,23,40)(25,43,31,45)(28,36,29,38)(32,35,48,41)(42,44,47,46)

Polycyclic presentation:

2^9|a^2=h,b^2=efi,c^2=ehi,d^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,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, 253227);