#234228

order	512
nilpotency class	3
isoclinism	isoclinic class of #234214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,12)(4,9)(5,10)(6,15)(7,11)(8,14)(13,16)(17,18)(19,27)(20,25)(21,24)(22,26)(23,28)(29,32)(30,31);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,12,10)(17,19,22,30)(18,23,26,32)(20,29,21,28)(24,31,25,27);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,10,16,15)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,12,16,9)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,31,32,27)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^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=dh,e^b=eh,f^b=fi,d^c=dh,e^c=eh,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, 141424);