#152870

order	512
nilpotency class	5
isoclinism	isoclinic class of #151066
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#367
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=ei,d^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dg,f^b=fi,g^b=gh,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60066);