#6458535

order	512
nilpotency class	3
isoclinism	isoclinic class of #6458534
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2504
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338519, #72338520, #72338524
maximal normal subgroup of	#72338519, #72338520, #72338524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365731);