#213927

order	512
nilpotency class	3
isoclinism	isoclinic class of #213876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#37187	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

962 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 962 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,g^c=gi,g^d=gh

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, 121123);