#6497699

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470867
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

1880 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)(21,22,27,30)(23,33,35,45)(24,38,40,48)(25,32,26,31)(28,37,44,36)(29,41,47,42)(34,43,39,46)(49,50,52,51);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,25,36,27,35,26,37)(22,28,31,33,30,44,32,45)(24,39,41,51,40,34,42,50)(29,46,48,52,47,43,38,49);(7,13)(10,17)(14,19)(18,20)(21,24,25,41,27,40,26,42)(22,29,31,48,30,47,32,38)(23,34,36,50,35,39,37,51)(28,43,33,49,44,46,45,52);(7,13)(10,17)(14,19)(18,20)(23,35)(24,40)(28,44)(29,47)(33,45)(36,37)(38,48)(41,42);(5,8,9,16)(6,11,12,15)(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|a^2=h,b^2=f,c^2=gi,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,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6404895);