#233624

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1214 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^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=dhi,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140820);