#147588

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1614 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)(7,17)(8,13)(9,26,28,14)(10,15)(11,16)(12,19,34,49)(18,43)(20,44)(21,45)(22,51)(23,53,55,36)(24,37)(25,38)(27,56,59,39)(29,60)(30,41)(31,42)(32,46,61,67)(33,48,40,57)(35,52,62,68)(47,65)(50,66)(54,64,58,63);(5,7,11,21)(6,12,16,34)(8,22,25,50)(9,19,28,49)(10,29,31,20)(13,35,38,62)(14,17,26,45)(15,40,42,33)(18,30,47,24)(23,52,55,68)(27,57,59,48)(32,41,61,37)(36,51,53,66)(39,60,56,44)(43,64,65,63)(46,58,67,54);(2,4)(5,8,11,25)(6,13,16,38)(7,18,21,47)(9,23,28,55)(10,30,31,24)(12,32,34,61)(14,36,26,53)(15,41,42,37)(17,43,45,65)(19,46,49,67)(20,50,29,22)(27,58,59,54)(33,62,40,35)(39,64,56,63)(44,66,60,51)(48,68,57,52)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,f^b=fi,g^b=gh,f^c=fi,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, 54784);