#488758

order	512
nilpotency class	4
isoclinism	isoclinic class of #488723
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	#28497	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 785 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) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395954);