#6496400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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)(23,28)(24,36)(25,32)(26,30)(27,31)(29,38)(33,49)(34,47)(35,44)(37,52)(39,48)(40,43)(41,46)(42,51)(45,50);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,28)(24,37)(25,40)(26,34)(27,35)(29,45)(30,47)(31,44)(32,43)(33,39)(36,52)(38,50)(41,51)(42,46)(48,49);(7,13)(10,17)(14,19)(18,20)(21,24,27,39)(22,29,31,46)(23,33,35,37)(25,38,26,41)(28,42,44,45)(30,36,32,48)(34,50,40,51)(43,52,47,49);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,32)(23,34,35,40)(24,38,39,41)(28,43,44,47)(29,36,46,48)(33,50,37,51)(42,52,45,49);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(23,35)(29,46)(30,32)(33,37)(34,40)(36,48)(50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403596);