#464654

order	512
nilpotency class	3
isoclinism	isoclinic class of #464654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16508
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1517 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215912, #72215972, #72216059, …
maximal normal subgroup of	#72215912, #72215972, #72216059, …
maximal quotient of	C₂×#464654, #72252120, #72314124

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371850);