#343064

order	512
nilpotency class	3
isoclinism	isoclinic class of #343051
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	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

681 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:

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

Polycyclic presentation:

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

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, 250260);