#187108

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	#62815	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2182 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:

(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(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);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(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);(2,7)(4,13)(6,9)(10,12)(17,20)(18,25)(19,29)(21,33)(22,37)(23,34)(24,38)(26,41)(27,32)(28,43)(30,44)(31,46)(35,42)(36,45)(39,48)(40,47);(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,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, 94304);