#108378

order	512
nilpotency class	3
isoclinism	isoclinic class of #108372
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

690 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)(3,13,17,22,16,23,15,8)(4,9,19,24,20,26,18,25)(14,28,29,21,30,27,31,32)(33,34)(35,44,48,59)(36,49)(37,42)(38,52)(39,43)(40,46,55,47)(41,50)(45,54,61,64)(51,56)(53,57)(58,62,63,60);(1,3,6,15,7,16,5,17)(2,8,10,23,12,22,11,13)(4,14,18,31,20,30,19,29)(9,21,25,28,26,32,24,27)(33,35)(34,40,43,55)(36,45)(37,46)(38,47)(39,48)(41,54,57,64)(42,44,52,59)(49,63,56,58)(50,60)(51,61)(53,62);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,21)(10,25)(11,24)(12,26)(13,27)(15,31)(16,30)(17,29)(22,32)(23,28)(33,36)(34,41)(35,45)(37,50)(38,53)(39,51)(40,54)(42,56)(43,57)(44,58)(46,60)(47,62)(48,61)(49,52)(55,64)(59,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=hi,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15574);