#6630693

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 conjugacy classes of subgroups; only the 570 normal subgroups are shown.

normal subgroups only

This group has 1626 conjugacy classes of subgroups — too many to draw. Only its 570 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 — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537889);