#493160

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) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

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,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	#72239482, #72239529, #72239550, …
maximal normal subgroup of	#72239482, #72239529, #72239550, …
maximal quotient of	#72319155, #72320019, C₂×#493160

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400356);