#6345908

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	#61433	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2965 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6253104);