#277854

order	512
nilpotency class	3
isoclinism	isoclinic class of #277851
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	#3100	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

600 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,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,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185050);