#113645

order	512
nilpotency class	3
isoclinism	isoclinic class of #113643
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#35329	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1198 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,g^b=ghi,d^c=di,e^c=ei,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318069, #72318390, #72318416
maximal normal subgroup of	#72318069, #72318390, #72318416

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20841);