#194579

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1084 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=hi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

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, 101775);