#151982

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	#2163
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 226 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4)(5,6)(7,17)(8,23)(9,15)(10,16)(11,13)(12,19)(14,24)(18,30)(20,31)(21,40)(22,38)(25,49)(26,50)(27,37)(28,33)(29,34)(32,45)(35,51)(36,41)(39,44)(42,56)(43,55)(46,54)(47,57)(48,60)(52,66)(53,58)(59,62)(61,64)(63,68)(65,67);(2,4)(5,7,8,20,9,21,25,47,10,22,26,46,27,48,52,65,11,19,24,44,28,45,51,64,29,18,41,42,53,43,62,63)(6,12,14,31,15,32,35,57,16,30,36,54,37,55,59,67,13,17,23,39,33,40,49,61,34,38,50,56,58,60,66,68);(6,13)(7,18,19,22)(8,24)(12,30,17,38)(15,33)(16,34)(20,46,44,42)(21,43,45,48)(25,51)(26,41)(31,56,39,54)(32,55,40,60)(37,58)(47,65,64,63)(52,62)(57,68,61,67)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59178);