#6345516

order	512
nilpotency class	3
isoclinism	isoclinic class of #61569
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1526 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:

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

Polycyclic presentation:

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

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, 6252712);