#6436663

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,g^c=gi

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, 6343859);