#6343391

order	512
nilpotency class	3
isoclinism	isoclinic class of #17597
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	#62788	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2835 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,d^c=dh

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250587);