#511547

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

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, 418743);