#109880

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17076);