#6345622

order	512
nilpotency class	3
isoclinism	isoclinic class of #62243
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	#60948	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1566 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)(4,8,13,16,14,11,12,15)(17,18)(19,28)(20,25)(21,26)(22,35)(23,27)(24,30)(29,43)(31,38)(32,40)(33,41)(34,42)(36,39)(37,45)(44,48)(46,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19)(18,24)(20,29)(21,30)(22,36)(23,31)(25,37)(26,38)(27,39)(28,35)(32,44)(33,45)(34,46)(40,43)(41,47)(42,48);(1,4,6,13,7,14,5,12)(2,8,10,16,3,11,9,15)(17,20)(18,25)(19,29)(21,33)(22,32)(23,34)(24,37)(26,41)(27,42)(28,43)(30,45)(31,46)(35,40)(36,44)(38,47)(39,48);(1,5,7,6)(2,9,3,10)(4,12,14,13)(8,15,11,16)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,34,33)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,43,48,47);(1,5,7,6)(2,10,3,9)(4,12,14,13)(8,16,11,15)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,33,34,32)(24,38,39,28)(25,41,42,40)(29,45,46,44)(37,47,48,43)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252818);