#187110

order	512
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94306);