#107231

order	512
nilpotency class	3
isoclinism	isoclinic class of #107214
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

726 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30)(33,34,37,42,38,41,51,56)(35,39,45,54,44,53,49,57)(36,46,47,58,48,40,50,55)(43,59,61,64,60,52,62,63);(1,3)(2,7,10,22)(4,12)(5,13)(6,14)(8,21,15,31)(9,11,18,26)(16,27)(17,28)(19,29)(20,30)(23,32,24,25)(33,35)(34,39,41,53)(36,43)(37,49)(38,44)(40,52,46,59)(42,57,56,54)(45,51)(47,62)(48,60)(50,61)(55,64,58,63);(1,4,6,17)(2,8,10,15)(3,12,14,28)(5,19,20,16)(7,21,22,31)(9,24,18,23)(11,25,26,32)(13,29,30,27)(33,36,38,48)(34,40,41,46)(35,43,44,60)(37,50,51,47)(39,52,53,59)(42,58,56,55)(45,62,49,61)(54,64,57,63)

Polycyclic presentation:

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

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, 14427);