#108111

order	512
nilpotency class	3
isoclinism	isoclinic class of #108094
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

724 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)(33,34,38,42,39,43,37,52)(35,44,48,59,46,40,47,55)(36,49,50,41,51,56,53,57)(45,60,62,58,61,64,63,54);(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)(34,40,43,44)(36,45)(37,47)(38,48)(39,46)(41,54,57,58)(42,55,52,59)(49,64,56,60)(50,62)(51,61)(53,63);(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)(34,41)(35,45)(37,53)(38,50)(39,51)(40,54)(42,56)(43,57)(44,58)(46,61)(47,63)(48,62)(49,52)(55,60)(59,64)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15307);