#213923

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	C₂×#535	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 980 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)(17,18,21,26,22,25,34,40)(19,27,30,41,29,23,33,36)(20,24,31,39,32,38,45,48)(28,35,43,46,42,37,44,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,19)(18,23)(20,28)(21,33)(22,29)(24,37)(25,27)(26,41)(30,34)(31,44)(32,42)(35,38)(36,40)(39,46)(43,45)(47,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20,22,32)(18,24,25,38)(19,28,29,42)(21,31,34,45)(23,35,27,37)(26,39,40,48)(30,43,33,44)(36,46,41,47);(3,10)(6,14)(9,16)(13,15)(18,25)(21,34)(23,27)(24,38)(30,33)(31,45)(35,37)(43,44)

Polycyclic presentation:

2^9|a^2=g,c^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, 121119);