#108676

order	512
nilpotency class	3
isoclinism	isoclinic class of #108672
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
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

415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi,e^c=ei,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, 15872);