#343109

order	512
nilpotency class	3
isoclinism	isoclinic class of #343099
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	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=efhi,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250305);