#268480

order	512
nilpotency class	3
isoclinism	isoclinic class of #268355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59135	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

(5,6,10,12,8,7,9,11)(13,14)(15,24)(16,21)(17,27)(18,28)(19,23)(20,25)(22,26);(1,2)(3,4)(9,10)(11,12)(13,15,19,17)(14,20,23,22)(16,25,18,26)(21,27,28,24);(2,4)(5,7,8,6)(9,11,10,12)(13,16,19,18)(14,21,23,28)(15,25,17,26)(20,27,22,24);(9,10)(11,12)(13,17,19,15)(14,22,23,20)(16,26,18,25)(21,24,28,27)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=h,d^2=i,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175676);