#285027

order	512
nilpotency class	3
isoclinism	isoclinic class of #284987
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#24770	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

696 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192223);