#6345127

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60676	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,d^b=di,f^b=fh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252323);