#129300

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2274 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

712 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36496);