#133492

order	512
nilpotency class	4
isoclinism	isoclinic class of #133422
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ #2309 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

672 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,e^b=eh,f^b=fg,d^c=dh,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303826
maximal normal subgroup of	#72303826
maximal quotient of	#72303980

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40688);