#493185

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

670 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=e,g^2=h,h^2=i,b^a=be,c^a=ci,g^a=gi,d^b=dh,e^b=ei,d^c=di,g^d=gi

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, 400381);