#187905

order	512
nilpotency class	3
isoclinism	isoclinic class of #187866
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1998 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,g^c=ghi

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, 95101);