#147806

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1023 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,6,11,17,12,18,10,16)(7,19,23,49,24,50,22,48)(8,25,29,14,30,42,28,43)(9,31,33,45,34,15,32,46)(13,35,38,21,39,55,37,56)(20,51,26,47,54,67,53,65)(27,60,62,41,63,58,61,44)(36,52,40,64,57,68,66,59);(5,7,12,24)(6,13,18,39)(8,26,30,53)(9,21,34,56)(10,22,11,23)(14,40,43,66)(15,19,31,50)(16,38,17,37)(20,29,54,28)(25,57,42,36)(27,59,63,64)(32,35,33,55)(41,67,44,51)(45,49,46,48)(47,60,65,58)(52,61,68,62);(2,4)(5,8,12,30)(6,14,18,43)(7,20,24,54)(9,27,34,63)(10,29,11,28)(13,36,39,57)(15,44,31,41)(16,42,17,25)(19,47,50,65)(21,52,56,68)(22,26,23,53)(32,62,33,61)(35,64,55,59)(37,40,38,66)(45,60,46,58)(48,67,49,51)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55002);