#108031

order	512
nilpotency class	3
isoclinism	isoclinic class of #108026
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	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

686 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)(3,11,13,7)(4,15,16,8)(5,10)(6,9)(12,23)(14,24,27,20)(17,28,29,21)(18,22)(19,25)(26,31)(30,32)(33,34)(35,39,44,54)(36,46)(37,49)(38,41)(40,47)(42,51)(43,57,58,64)(45,61,60,53)(48,56)(50,55)(52,62,63,59);(1,3,5,14,6,13,18,27)(2,7,10,20,9,11,22,24)(4,12,17,26,16,25,29,32)(8,19,21,30,15,23,28,31)(33,35)(34,39,41,54)(36,43)(37,45)(38,44)(40,52,55,63)(42,53,49,61)(46,57,56,64)(47,59)(48,58)(50,62)(51,60);(1,4,6,16)(2,8,9,15)(3,12,13,25)(5,17,18,29)(7,19,11,23)(10,21,22,28)(14,26,27,32)(20,30,24,31)(33,36,38,48)(34,40,41,55)(35,43,44,58)(37,50,51,47)(39,52,54,63)(42,56,49,46)(45,62,60,59)(53,64,61,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,d^c=dhi,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, 15227);