#496095

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3147	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 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)(5,6,10,16,11,15,21,31,12,17,20,13,9,26,7,18)(8,14,24,30,25,29,33,36,22,28,34,27,23,35,19,32);(5,7,9,20,12,21,11,10)(6,13,15,18,17,16,26,31)(8,19,23,34,22,33,25,24)(14,27,29,32,28,30,35,36);(1,2,3,4)(8,22)(14,28)(19,33)(23,25)(24,34)(27,30)(29,35)(32,36);(5,8,10,24,11,25,21,33,12,22,20,34,9,23,7,19)(6,14,16,30,15,29,31,36,17,28,13,27,26,35,18,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=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, 403291);