#6458777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442595
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2482
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2187 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249829, #72306445, #72306460, …
maximal normal subgroup of	#72249829, #72306445, #72306460, …
maximal quotient of	#72257147

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365973);