#332000

order	512
nilpotency class	3
isoclinism	isoclinic class of #331999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10212	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

525 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 525 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239196);