#332003

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

Polycyclic presentation:

2^9|c^2=h,d^2=i,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, 239199);