#444083

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15772
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

801 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)(3,13,17,28)(4,18,23,20)(5,7)(6,11)(8,12)(9,15,25,16)(10,21,27,22)(14,24)(19,26)(29,32)(30,31);(1,3,7,15,8,17,6,16)(2,9,5,13,12,25,11,28)(4,19,21,14,23,29,22,30)(10,26,18,24,27,32,20,31);(1,4)(2,10)(3,14,17,30)(5,20)(6,21)(7,22)(8,23)(9,24,25,31)(11,18)(12,27)(13,26,28,32)(15,19,16,29);(1,5,6,2,8,11,7,12)(3,9,15,13,17,25,16,28)(4,20,21,10,23,18,22,27)(14,24,19,26,30,31,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338512, #72338513, #72338524, …
maximal normal subgroup of	#72338512, #72338513, #72338524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351279);