#455703

order	512
nilpotency class	3
isoclinism	isoclinic class of #453999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(2,5)(6,8)(9,10,15,20)(11,21,25,46)(12,26,29,47)(13,18,32,45)(14,33,37,19)(16,34,40,24)(17,35,42,28)(22,38,53,67)(23,54,57,50)(27,60,61,43)(30,63,62,41)(31,48,36,44)(39,59,69,56)(49,58,66,52)(51,71,72,65)(55,68,64,70);(1,2,4,5)(3,6,7,8)(9,11,15,25)(10,16,20,40)(12,22,29,53)(13,23,32,57)(14,34,37,24)(17,38,42,67)(18,39,45,69)(19,46,33,21)(26,49,47,66)(27,51,61,72)(28,58,35,52)(30,64,62,55)(31,59,36,56)(41,65,63,71)(43,70,60,68)(44,50,48,54);(1,3)(2,6)(4,7)(5,8)(9,12,15,29)(10,17,20,42)(11,22,25,53)(13,30,32,62)(14,35,37,28)(16,38,40,67)(18,43,45,60)(19,47,33,26)(21,49,46,66)(23,55,57,64)(24,58,34,52)(27,36,61,31)(39,68,69,70)(41,48,63,44)(50,71,54,65)(51,59,72,56);(3,7)(6,8)(9,13)(10,18)(11,23)(12,27)(14,36)(15,32)(16,39)(17,41)(19,48)(20,45)(21,50)(22,51)(24,59)(25,57)(26,60)(28,62)(29,61)(30,35)(31,37)(33,44)(34,56)(38,65)(40,69)(42,63)(43,47)(46,54)(49,68)(52,64)(53,72)(55,58)(66,70)(67,71)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=i,e^2=i,f^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,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 362899);