#147629

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#481
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1058 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)(9,10,13,18,14,17,29,19)(11,20,22,33,23,36,28,37)(12,24,27,16,26,35,40,34)(15,25,31,21,30,39,32,38);(5,6)(7,8)(9,11,14,23)(10,15,17,30)(12,25,26,39)(13,28,29,22)(16,33,34,37)(18,32,19,31)(20,24,36,35)(21,27,38,40);(2,4)(6,8)(12,26)(15,30)(16,34)(21,38)(24,35)(25,39)(27,40)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282164, #72282251, #72282356
maximal normal subgroup of	#72282164, #72282251, #72282356

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54825);