#493158

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#442
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

726 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,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,e^c=ei,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239482, #72239532, #72239540, …
maximal normal subgroup of	#72239482, #72239532, #72239540, …
maximal quotient of	#72285908, #72309649, #72310155, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400354);