#123841

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1442
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

1434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237176, #72237180, #72237189, …
maximal normal subgroup of	#72237176, #72237180, #72237189, …
maximal quotient of	#72239045

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31037);