#128013

order	512
nilpotency class	4
isoclinism	isoclinic class of #128009
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂⁴×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7)(4,6)(5,8)(9,10,13,17,14,18,29,36)(11,19,22,37,20,33,27,39)(12,23,25,35,26,16,28,34)(15,30,32,24,31,38,40,21);(1,3)(2,5,6,7)(4,8)(9,11)(10,15)(12,24)(13,27)(14,20)(16,33)(17,32)(18,31)(19,23)(21,26)(22,29)(25,30)(28,38)(34,39)(35,37)(36,40);(11,20)(15,31)(19,33)(21,24)(22,27)(30,38)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gi,c^b=ci,d^b=dh,e^b=egi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267702
maximal normal subgroup of	#72267702
maximal quotient of	#72267712

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35209);