#388088

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1015 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=efg,d^2=efg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295284);