#233603

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#36687	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303167, #72303170, #72303288, …
maximal normal subgroup of	#72303167, #72303170, #72303288, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140799);