#6376359

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283555);