#272084

order	512
nilpotency class	3
isoclinism	isoclinic class of #272027
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	#32921	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

924 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)(9,10,13,23,15,12,14,17)(11,18,20,22,21,24,19,16);(2,3)(7,8)(9,11,15,21)(10,16,12,22)(13,20,14,19)(17,24,23,18);(1,3,4,2)(5,7,6,8)(9,12,15,10)(11,18,21,24)(13,23,14,17)(16,19,22,20);(9,10,15,12)(11,16,21,22)(13,23,14,17)(18,19,24,20)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=h,e^2=h,f^2=i,g^2=h,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=gh,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, 179280);