#122185

order	512
nilpotency class	3
isoclinism	isoclinic class of #97690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22)(25,26)(27,35)(28,39,41,32)(29,42)(30,34)(31,37)(33,45)(36,51,53,56)(38,49)(40,48,44,50)(43,47)(46,52,55,54);(1,2,4,6)(3,7,8,5)(9,11,12,18)(10,14,15,17)(13,21,20,19)(16,24,23,22)(25,27)(26,31)(28,36)(29,43)(30,38)(32,46)(33,49)(34,47)(35,42)(37,45)(39,55)(40,54)(41,53)(44,52)(48,51)(50,56);(1,3,4,8)(2,5,6,7)(10,15)(11,18)(13,20)(22,24)(25,28,30,41)(26,32,34,39)(27,36,38,53)(29,44,45,40)(31,46,47,55)(33,50,42,48)(35,51,49,56)(37,54,43,52)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29381);