#343078

order	512
nilpotency class	3
isoclinism	isoclinic class of #343075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

754 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defhi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250274);