#108324

order	512
nilpotency class	3
isoclinism	isoclinic class of #108316
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

706 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:

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

Polycyclic presentation:

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

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, 15520);