#455821

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
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

1064 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)(5,6,11,16)(7,17,20,12)(8,22,25,38)(9,14,27,37)(10,28,30,15)(13,29,35,24)(18,42,44,51)(19,40,47,32)(21,33,49,41)(23,52,54,62)(26,55,57,36)(31,45,39,50)(34,56,61,53)(43,64,46,66)(48,60,68,63)(58,65,59,67);(5,7,10,21,11,20,30,49)(6,12,15,33,16,17,28,41)(8,18,24,45,25,44,29,50)(9,19,26,48,27,47,57,68)(13,31,22,51,35,39,38,42)(14,32,36,60,37,40,55,63)(23,43,53,65,54,46,56,67)(34,58,52,66,61,59,62,64);(5,8)(6,13)(7,18,20,44)(9,23)(10,29)(11,25)(12,31,17,39)(14,34)(15,38)(16,35)(19,46,47,43)(21,50,49,45)(22,28)(24,30)(26,56)(27,54)(32,59,40,58)(33,42,41,51)(36,62)(37,61)(48,65,68,67)(52,55)(53,57)(60,66,63,64);(2,4)(5,9,11,27)(6,14,16,37)(7,19,20,47)(8,23,25,54)(10,26,30,57)(12,32,17,40)(13,34,35,61)(15,36,28,55)(18,43,44,46)(21,48,49,68)(22,52,38,62)(24,53,29,56)(31,58,39,59)(33,60,41,63)(42,64,51,66)(45,65,50,67)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 363017);