#333148

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
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

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

normal subgroups only

This group has 605 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:

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

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240344);