#149303

order	512
nilpotency class	3
isoclinism	isoclinic class of #8736
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

490 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

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, 56499);