#267651

order	512
nilpotency class	3
isoclinism	isoclinic class of #267563
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	#25754	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

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, 174847);