#405850

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14919
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319325, #72319352, #72319444
maximal normal subgroup of	#72319325, #72319352, #72319444

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313046);