#493506

order	512
nilpotency class	4
isoclinism	isoclinic class of #493500
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62876	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3500 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2277 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)(3,13)(4,16,18,27)(5,19,21,10)(6,11)(7,12)(8,15,24,20)(9,14,25,22)(17,30)(23,31)(26,28)(29,32);(1,3,7,6)(2,8,12,24)(4,14,18,22)(5,15,21,20)(9,23,25,30)(10,13,19,11)(16,29,27,26)(17,28,31,32);(1,4,7,18)(2,9,12,25)(3,14,6,22)(5,17,21,31)(8,23,24,30)(10,26,19,29)(11,27,13,16)(15,28,20,32);(3,6)(8,24)(9,25)(11,13)(14,22)(15,20)(26,29)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232878, #72260481, #72260534
maximal normal subgroup of	#72232878, #72260481, #72260534
maximal quotient of	#72232751, #72233316, #72237394, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400702);