#320216

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

817 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)(17,18,20,24,21,23,29,31)(19,25,27,30,26,32,28,22);(2,5)(6,7)(9,10,11,14,12,15,16,13)(19,26)(22,30)(25,32)(27,28);(10,13)(11,16)(14,15)(17,19,21,26)(18,22,23,30)(20,28,29,27)(24,32,31,25);(1,3,4,8)(2,6,5,7)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=h,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, 227412);