#6396759

order	512
nilpotency class	3
isoclinism	isoclinic class of #62295
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	C₂²×C₄	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 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1599 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26,23,27,21,37)(19,24,30,39,31,40,29,46)(20,32,36,42,34,48,35,25)(28,43,45,41,33,47,44,38);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,30,23,31,21,29)(18,24,26,39,27,40,37,46)(20,33,36,44,34,28,35,45)(25,41,32,47,42,38,48,43);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20)(18,25,27,42)(19,28)(21,35)(22,36)(23,34)(24,38,40,41)(26,32,37,48)(29,44)(30,45)(31,33)(39,43,46,47);(3,12)(7,10)(11,13)(15,16)(20,34)(25,42)(28,33)(32,48)(35,36)(38,41)(43,47)(44,45);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(20,34)(25,42)(28,33)(32,48)(35,36)(38,41)(43,47)(44,45)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fgh,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,d^c=di,e^c=ei

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, 6303955);