#493127

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=h,h^2=i,b^a=be,d^a=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400323);