#396108

order	512
nilpotency class	3
isoclinism	isoclinic class of #396107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1479 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:

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

Polycyclic presentation:

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

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, 303304);