#500018

order	512
nilpotency class	4
isoclinism	isoclinic class of #499849
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1562 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1562 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340589, #72340621, #72340706
maximal normal subgroup of	#72340589, #72340621, #72340706

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407214);