#6345570

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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	#60944	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1529 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)(4,8,12,15,13,11,14,16)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,31)(30,43)(32,42)(33,40)(34,41)(35,39)(36,38)(37,46)(44,47)(45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,13)(12,16)(14,15)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(1,4,6,12,7,13,5,14)(2,8,10,15,3,11,9,16)(17,20)(18,25)(19,30)(21,35)(22,34)(23,33)(24,37)(26,40)(27,41)(28,39)(29,43)(31,46)(32,44)(36,45)(38,48)(42,47);(2,3)(8,11)(9,10)(15,16)(18,26)(20,33)(24,38)(27,28)(29,42)(30,44)(34,35)(45,46);(1,5,7,6)(2,9,3,10)(4,12,13,14)(8,15,11,16)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,39,40,41)(30,45,44,46)(37,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252766);