#494185

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401381);