#492926

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

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

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, 400122);