#147858

order	512
nilpotency class	4
isoclinism	isoclinic class of #13432
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

1019 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,f^2=i,g^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,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, 55054);