#6345845

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
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	#61430	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2922 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1546 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)(20,25)(21,27)(22,35)(23,26)(24,30)(29,42)(31,39)(32,41)(33,40)(34,45)(36,38)(37,44)(43,47)(46,48);(17,19,23,31)(18,24,26,38)(20,29,32,43)(21,30,22,36)(25,37,41,48)(27,39,35,28)(33,44,34,46)(40,47,45,42);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(17,20,23,32)(18,25,26,41)(19,29,31,43)(21,33,22,34)(24,37,38,48)(27,40,35,45)(28,42,39,47)(30,44,36,46);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(18,26)(20,32)(24,38)(27,35)(28,39)(29,43)(33,34)(44,46);(4,12)(8,10)(13,14)(15,16)(17,21,23,22)(18,27,26,35)(19,30,31,36)(20,33,32,34)(24,39,38,28)(25,40,41,45)(29,44,43,46)(37,47,48,42)

Polycyclic presentation:

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

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, 6253041);