#494778

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#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×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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)(3,4)(5,6,10,7,9,14,25,18,11,15,28,19,26,13,33,17)(8,20,24,34,23,30,36,16,22,31,27,21,35,12,29,32);(5,7,10,14,9,18,25,15,11,19,28,13,26,17,33,6)(8,21,24,12,23,32,36,20,22,34,27,30,35,16,29,31);(2,4)(5,8,9,23,11,22,26,35)(6,12,13,31,15,30,14,20)(7,16,17,34,19,32,18,21)(10,27,25,29,28,24,33,36);(8,22)(12,30)(16,32)(20,31)(21,34)(23,35)(24,27)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=ch,e^b=ei,d^c=di,g^c=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, 401974);