#187858

order	512
nilpotency class	3
isoclinism	isoclinic class of #187780
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3547	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2153 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:

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

Polycyclic presentation:

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

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, 95054);