#6432446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂²×#400
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1871 conjugacy classes of subgroups; only the 824 normal subgroups are shown.

normal subgroups only

This group has 1871 conjugacy classes of subgroups — too many to draw. Only its 824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307021, #72307055, #72319030, …
maximal normal subgroup of	#72307021, #72307055, #72319030, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339642);