#346134

order	512
nilpotency class	3
isoclinism	isoclinic class of #346096
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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

503 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)(3,10)(4,13)(5,7)(6,9)(8,12)(11,15)(14,16)(17,18,21,27)(19,24,33,42)(20,34,35,44)(22,29,23,28)(25,38,26,36)(30,40,47,43)(31,48,46,39)(32,45,37,41);(2,7)(3,11)(4,6)(8,15)(10,16)(12,14)(17,19,23,32,21,33,22,37)(18,24,29,41,27,42,28,45)(20,31,38,30,35,46,36,47)(25,43,44,39,26,40,34,48);(1,3,5,12)(2,8,7,10)(4,11,6,14)(9,15,13,16)(17,20)(18,25)(19,30,33,47)(21,35)(22,38)(23,36)(24,39,42,48)(26,27)(28,44)(29,34)(31,37,46,32)(40,41,43,45);(1,4,5,6)(2,9,7,13)(3,11,12,14)(8,15,10,16)(17,20)(18,26)(19,31)(21,35)(22,38)(23,36)(24,40)(25,27)(28,34)(29,44)(30,37)(32,47)(33,46)(39,41)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=eh,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=defi,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

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253330);