#107875

order	512
nilpotency class	3
isoclinism	isoclinic class of #107858
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

701 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)(33,34,38,43,39,44,37,42)(35,45,48,40,49,55,47,56)(36,50,51,41,53,57,52,58)(46,60,61,64,62,54,63,59);(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)(36,46)(37,48)(38,47)(39,49)(41,54)(42,55)(43,45)(44,56)(50,59)(51,63)(52,61)(53,62)(57,64)(58,60);(1,4)(2,9)(3,14)(5,19)(6,18)(7,20)(8,21)(10,26)(11,24)(12,25)(13,27)(15,31)(16,30)(17,29)(22,32)(23,28)(33,36)(34,41)(35,46)(37,51)(38,52)(39,53)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60)

Polycyclic presentation:

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

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, 15071);