#470931

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16717
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 378127);