#277900

order	512
nilpotency class	3
isoclinism	isoclinic class of #277891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

597 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185096);