#123602

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237177, #72237178, #72237179, …
maximal normal subgroup of	#72237177, #72237178, #72237179, …
maximal quotient of	#72239060, C₂×#123602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30798);