#6508915

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60753	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

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, 6416111);