#127134

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#7928
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,12,7,15,29,45,8,14,6,13,28,44,30,46)(3,16,19,48,21,37,55,61,22,50,20,49,54,32,56,60)(4,23,24,43,25,39,57,42,26,11,27,40,58,10,36,38)(9,31,33,51,35,59,62,18,34,52,41,53,47,64,63,17);(1,3,6,20,7,21,30,56,8,22,5,19,28,54,29,55)(2,9,13,33,15,35,46,62,14,34,12,41,44,47,45,63)(4,18,24,52,25,53,57,64,26,17,27,31,58,51,36,59)(10,37,38,60,23,50,43,48,39,32,42,61,11,16,40,49);(1,4)(2,10,14,39)(3,17)(5,27)(6,24)(7,25)(8,26)(9,32,34,37)(11,15,23,44)(12,42,13,38)(16,47,50,35)(18,22)(19,52)(20,31)(21,51)(28,58)(29,36)(30,57)(33,61,41,60)(40,46,43,45)(48,62,49,63)(53,54)(55,64)(56,59)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ch,d^a=dg,g^a=gi,c^b=ci,e^b=eg,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286336, #72287445, #72287453, …
maximal normal subgroup of	#72286336, #72287445, #72287453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34330);