#465629

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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:

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

Polycyclic presentation:

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

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, 372825);