#6433175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433173
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2622
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60369	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2303 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338484, #72338487, #72338492
maximal normal subgroup of	#72338484, #72338487, #72338492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340371);