#343108

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	#3548	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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=h,c^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, 250304);