#454693

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,3,4)(5,6)(7,19)(8,25)(9,15)(10,17)(11,16)(12,18)(13,22)(14,29)(20,49)(21,53)(23,38)(24,37)(26,47)(27,43)(28,42)(30,41)(31,40)(32,45)(33,44)(34,46)(35,52)(36,55)(39,51)(48,57)(50,60)(54,63)(56,62)(58,65)(59,64)(61,67)(66,68);(5,7)(6,13)(8,26)(9,21)(10,23)(11,22)(12,24)(14,39)(15,36)(16,37)(17,19)(18,38)(20,29)(25,49)(27,54)(28,52)(30,51)(31,50)(32,56)(33,55)(34,57)(35,41)(40,63)(42,47)(43,66)(44,48)(45,53)(46,62)(58,68)(59,67)(60,64)(61,65);(2,4)(5,8,12,30)(6,14,18,42)(7,20,24,52)(9,31,34,58)(10,29,11,28)(13,35,38,49)(15,43,46,64)(16,25,17,41)(19,47,37,39)(21,54,57,67)(22,26,23,51)(27,33,59,32)(36,61,62,63)(40,45,65,44)(48,66,53,60)(50,56,68,55);(5,9,12,34)(6,15,18,46)(7,21,24,57)(8,27,30,59)(10,33,11,32)(13,36,38,62)(14,40,42,65)(16,45,17,44)(19,48,37,53)(20,50,52,68)(22,56,23,55)(25,43,41,64)(26,54,51,67)(28,58,29,31)(35,60,49,66)(39,63,47,61)

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=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, 361889);