#6498843

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei

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, 6406039);