#495077

order	512
nilpotency class	4
isoclinism	isoclinic class of #2230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

688 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224792, #72224799, #72226755, …
maximal normal subgroup of	#72224792, #72224799, #72226755, …
maximal quotient of	C₄×#13186

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402273);