#493773

order	512
nilpotency class	4
isoclinism	isoclinic class of #481284
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1098 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)(7,15,16,32)(8,18)(9,21)(11,23,24,35)(13,28)(14,27)(17,19,22,33)(20,31)(25,29,30,36)(26,34);(1,2,3,4)(5,6)(7,11)(8,13)(9,14)(10,12)(15,23)(16,24)(17,25)(18,28)(19,29)(20,26)(21,27)(22,30)(31,34)(32,35)(33,36);(5,7)(6,11)(8,19)(9,22)(10,16)(12,24)(13,29)(14,30)(15,31)(17,21)(18,32)(20,33)(23,34)(25,27)(26,36)(28,35);(6,12)(11,24)(13,26)(14,27)(23,35)(25,30)(28,34)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239492, #72250341, #72250367, …
maximal normal subgroup of	#72239492, #72250341, #72250367, …
maximal quotient of	#72319137, #72320010

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400969);