#396109

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1455 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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.

Relationships

maximal subgroup of	#72319996
maximal normal subgroup of	#72319996

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303305);