#320238

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

797 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)(17,18,20,25,21,24,28,23)(19,22,27,29,26,30,32,31);(2,5)(6,7)(9,10,13,14,12,16,11,15)(17,19)(18,22)(20,27)(21,26)(23,31)(24,30)(25,29)(28,32);(10,14)(11,13)(15,16)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(1,3,4,8)(2,6,5,7)(18,24)(22,30)(23,25)(29,31)

Polycyclic presentation:

2^9|a^2=f,b^2=ghi,d^2=e,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	#72224717, #72224721, #72224723
maximal normal subgroup of	#72224717, #72224721, #72224723

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 227434);