#219479

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126675);