#6468118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447057
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2571
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1027 conjugacy classes of subgroups; only the 568 normal subgroups are shown.

normal subgroups only

This group has 1027 conjugacy classes of subgroups — too many to draw. Only its 568 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,7,12)(3,13)(4,10,19,27)(5,11,6,9)(8,16)(14,24)(15,29)(17,23)(18,22)(20,28,21,25)(26,31)(30,32);(1,3)(2,8,12,24)(4,18)(5,14)(6,16)(7,17)(9,23,11,13)(10,26,27,29)(15,20)(19,30)(21,31)(22,25,32,28);(2,9,12,11)(3,14,17,16)(10,25,27,28)(15,30,31,18);(1,4,5,20,7,19,6,21)(2,10,11,28,12,27,9,25)(3,15,16,18,17,31,14,30)(8,22,23,26,24,32,13,29);(4,19)(10,27)(15,31)(18,30)(20,21)(22,32)(25,28)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375314);