#417815

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35676	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2812 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 2812 conjugacy classes of subgroups — too many to draw. Only its 258 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,17)(13,15)(14,25)(18,24)(19,30)(20,31)(21,28)(22,29)(23,34)(26,32)(27,33)(35,39)(36,40)(37,38);(10,15)(11,19,21,26)(16,30,28,32)(17,29)(18,25)(20,36,37,35)(31,40,38,39)(33,34);(2,4)(9,11)(10,16)(12,20)(13,21)(14,26)(15,28)(17,31)(18,30)(19,24)(22,37)(23,35)(25,32)(27,36)(29,38)(33,40)(34,39);(6,8)(9,12,13,22)(10,17,15,29)(11,20,21,37)(14,27,24,23)(16,31,28,38)(18,34,25,33)(19,35,26,36)(30,39,32,40)

Polycyclic presentation:

2^9|b^2=ei,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325011);