#147801

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
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

1039 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1039 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)(7,17,20,40)(8,13)(9,26,28,14)(10,15)(11,16)(12,19)(18,48,50,46)(21,47,30,63)(22,53,52,66)(23,41,56,37)(24,38)(25,39)(27,35)(29,58,61,42)(31,44)(32,45)(33,49)(34,51)(36,54)(43,60)(55,65)(57,68,62,67)(59,64);(5,7,11,20)(6,12,16,35)(8,22,25,52)(9,27,28,19)(10,30,32,21)(13,36,39,65)(14,40,26,17)(15,43,45,34)(18,31,50,24)(23,55,56,54)(29,51,61,60)(33,44,64,38)(37,66,41,53)(42,47,58,63)(46,68,48,67)(49,57,59,62);(2,4)(5,8,11,25)(6,13,16,39)(7,18,20,50)(9,23,28,56)(10,31,32,24)(12,33,35,64)(14,41,26,37)(15,44,45,38)(17,46,40,48)(19,49,27,59)(21,52,30,22)(29,62,61,57)(34,65,43,36)(42,67,58,68)(47,53,63,66)(51,55,60,54)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,e^b=ehi,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,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, 54997);