#108097

order	512
nilpotency class	3
isoclinism	isoclinic class of #108094
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

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	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,6,8)(3,5,10,14)(4,7)(9,11)(12,16)(13,15)(17,18)(19,27)(20,31)(21,34)(22,26)(23,29)(24,32)(25,36)(28,43)(30,38)(33,40)(35,39)(37,47)(41,45)(42,46)(44,48);(1,3,5,8,6,10,14,2)(4,9,13,16,12,15,11,7)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,37,39,41)(25,38,34,42)(31,48,40,43)(32,45)(33,44)(35,47)(36,46);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,20,22,33)(18,24,26,39)(19,28,29,44)(21,35,36,32)(23,37,27,41)(25,40,34,31)(30,47,46,45)(38,43,42,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=i,g^2=h,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, 15293);