#493454

order	512
nilpotency class	4
isoclinism	isoclinic class of #493451
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1782
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62855	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3428 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261390, #72261408, #72261527
maximal normal subgroup of	#72261390, #72261408, #72261527

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400650);