#123620

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238861
maximal normal subgroup of	#72238861
maximal quotient of	#72238868, #72238870, #72238876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30816);