#320318

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,15,11,14,12,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(2,5)(6,7)(10,14)(15,16)(17,19,21,28,20,27,29,26)(18,22,24,31,23,25,32,30);(10,15)(12,13)(14,16)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28);(1,3,4,8)(2,6,5,7)(18,23)(22,25)(24,32)(30,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224718, #72224721, #72224725
maximal normal subgroup of	#72224718, #72224721, #72224725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227514);