#405847

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
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

945 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)(3,13,17,29)(4,9)(5,20,22,10)(6,11)(7,12)(8,16,25,32)(14,30,27,21)(15,19)(18,26)(23,28)(24,31);(1,3)(2,8,12,25)(4,14,19,27)(5,15)(6,16)(7,17)(9,22)(10,24,20,18)(11,13,28,29)(21,31)(23,32)(26,30);(1,4,6,18,7,19,23,24)(2,9,11,26,12,15,28,31)(3,14,16,10,17,27,32,20)(5,13,21,25,22,29,30,8);(1,5)(2,10)(3,15)(4,13)(6,21)(7,22)(8,24)(9,17)(11,27)(12,20)(14,28)(16,31)(18,25)(19,29)(23,30)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338511, #72338517, #72338528
maximal normal subgroup of	#72338511, #72338517, #72338528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313043);