#343125

order	512
nilpotency class	3
isoclinism	isoclinic class of #343112
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3548	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

623 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 250321);