#493159

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319775, #72319789, C₄×#33471, …
maximal normal subgroup of	#72319775, #72319789, C₄×#33471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400355);