#512158

order	512
nilpotency class	4
isoclinism	isoclinic class of #511839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3027
commutator, Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	#33021
prime radical	#23994	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^a=hi,f^c=fi,g^c=ghi,f^d=fh,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343000
maximal normal subgroup of	#72343000

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419354);