#134661

order	512
nilpotency class	4
isoclinism	isoclinic class of #12833
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216392, #72216444, #72216451, …
maximal normal subgroup of	#72216392, #72216444, #72216451, …
maximal quotient of	#72227108

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41857);