#147809

order	512
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
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

1017 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1017 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)(7,18)(8,24)(9,30,32,15)(10,33)(11,16)(12,17)(13,20,38,54)(14,27)(19,49)(21,55)(22,47)(23,48)(25,58)(26,60,62,57)(28,41)(29,45)(31,43,34,44)(35,59,65,68)(36,56,37,53)(39,66,64,50)(40,63,42,61)(46,51)(52,67);(5,7,12,23)(6,13,17,38)(8,25,29,51)(9,20,32,54)(10,22,11,21)(14,39,41,64)(15,18,30,48)(16,36,33,37)(19,27,52,28)(24,35,45,65)(26,59,62,68)(31,56,34,53)(40,67,42,49)(43,47,44,55)(46,57,58,60)(50,63,66,61);(2,4)(5,8,12,29)(6,14,17,41)(7,19,23,52)(9,26,32,62)(10,28,11,27)(13,35,38,65)(15,42,30,40)(16,45,33,24)(18,46,48,58)(20,50,54,66)(21,25,22,51)(31,63,34,61)(36,39,37,64)(43,60,44,57)(47,67,55,49)(53,68,56,59)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=dh,e^c=ei,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, 55005);