#6497693

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1894 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,33)(24,38)(25,30)(26,32)(27,31)(28,37)(29,41)(34,43)(35,45)(36,44)(39,46)(40,47)(42,48)(49,50)(51,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,37,27,36,26,35)(22,28,32,33,31,45,30,44)(24,39,40,50,42,34,41,51)(29,46,48,52,47,43,38,49);(7,13)(10,17)(14,19)(18,20)(21,24,26,41,27,42,25,40)(22,29,30,38,31,47,32,48)(23,34,35,50,36,39,37,51)(28,43,44,52,45,46,33,49);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,32)(23,35,36,37)(24,40,42,41)(28,33,45,44)(29,38,47,48)(34,50,39,51)(43,49,46,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(23,36)(29,47)(30,32)(34,39)(35,37)(38,48)(50,51)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,d^2=i,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, 6404889);