#146400

order	512
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

886 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2,3,4)(5,6,9,15,11,17,29,41,12,18,10,16,28,40,30,42)(7,13,21,33,23,35,49,58,20,32,22,34,48,57,50,59)(8,14,25,37,27,39,52,61,24,36,26,38,51,60,53,62)(19,43,45,55,47,63,65,68,44,31,46,54,64,56,66,67);(5,7,10,22,11,23,30,50,12,20,9,21,28,48,29,49)(6,13,16,34,17,35,42,59,18,32,15,33,40,57,41,58)(8,19,26,46,27,47,53,66,24,44,25,45,51,64,52,65)(14,31,38,55,39,56,62,68,36,43,37,54,60,63,61,67);(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,24)(13,31)(15,37)(16,38)(17,39)(18,36)(20,44)(21,45)(22,46)(23,47)(28,51)(29,52)(30,53)(32,43)(33,54)(34,55)(35,56)(40,60)(41,61)(42,62)(48,64)(49,65)(50,66)(57,63)(58,67)(59,68)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53596);