#114048

order	512
nilpotency class	3
isoclinism	isoclinic class of #114044
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	C₂×#535	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1068 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 142 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)(11,21,25,31)(12,26,28,35)(13,18)(14,19)(15,20)(16,23,32,24)(17,27,34,29)(22,33)(30,38)(36,39)(37,40);(2,5)(3,7)(9,11)(10,16)(12,22)(13,24)(14,23)(15,25)(17,33)(18,31)(19,21)(20,32)(26,36)(27,39)(28,38)(29,37)(30,34)(35,40);(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,26)(19,35)(20,34)(21,36)(23,37)(24,39)(25,38)(31,40)(32,33)

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,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,e^c=ei,g^c=gi

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, 21244);