#451175

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15987
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

769 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)(3,14,17,8)(4,18)(5,12)(6,13)(7,11)(9,20)(10,16,28,23)(15,24,31,30)(19,25,32,26)(21,29)(22,27);(1,3)(2,8)(4,19,20,32)(5,16)(6,23)(7,17)(9,25,18,26)(10,12)(11,14)(13,28)(15,21,31,22)(24,27,30,29);(1,4)(2,9,11,18)(3,15)(5,22)(6,21)(7,20)(8,24,14,30)(10,26,28,25)(12,29,13,27)(16,32)(17,31)(19,23);(1,3)(2,10)(4,15)(5,16)(6,23)(7,17)(8,13)(9,26)(11,28)(12,14)(18,25)(19,21)(20,31)(22,32)(24,27)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319162, #72319325, #72319370, …
maximal normal subgroup of	#72319162, #72319325, #72319370, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358371);