#185396

order	512
nilpotency class	3
isoclinism	isoclinic class of #185249
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	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2173 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:

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

Polycyclic presentation:

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

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, 92592);