#508027

order	512
nilpotency class	4
isoclinism	isoclinic class of #508016
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32942	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

991 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 991 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415223);