#123529

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2454 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)(9,10,13,18,14,17,27,32)(11,15,21,30,20,29,25,33)(12,22,23,34,24,16,26,31)(19,35,37,40,36,28,38,39);(5,6)(7,8)(9,11)(10,15,17,29)(12,19)(13,25)(14,20)(16,28,22,35)(18,33,32,30)(21,27)(23,38)(24,36)(26,37)(31,40,34,39);(2,4)(6,8)(9,12,14,24)(10,16,17,22)(11,19,20,36)(13,26,27,23)(15,28,29,35)(18,34,32,31)(21,38,25,37)(30,40,33,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30725);