#234231

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

1603 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1603 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,20)(12,17)(13,18)(14,23)(15,19)(16,22)(21,24);(2,6)(3,4)(7,8)(9,11,15,12)(10,16,19,24)(13,22,14,21)(17,23,20,18);(2,7)(3,4)(6,8)(9,12,15,11)(10,17,19,20)(13,21,14,22)(16,18,24,23);(1,3,5,4)(2,6,8,7)(9,13,15,14)(10,18,19,23)(11,21,12,22)(16,20,24,17)

Polycyclic presentation:

2^9|b^2=hi,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, 141427);