#285109

order	512
nilpotency class	3
isoclinism	isoclinic class of #285051
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	#18624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=i,d^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,d^b=di,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, 192305);