#272325

order	512
nilpotency class	3
isoclinism	isoclinic class of #272283
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61070	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1069 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,f^c=fi,g^c=gi,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, 179521);