#6461516

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×#2598
prime radical	#23434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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)(3,12)(4,9)(5,20)(6,11)(7,10)(8,14)(13,29)(15,23)(16,24)(17,22)(18,28)(19,26)(21,27)(25,31)(30,32);(1,3)(2,8)(4,17)(5,16)(6,14)(7,15)(9,25)(10,24)(11,23)(12,20)(13,18)(19,30)(21,31)(22,28)(26,29)(27,32);(1,4,6,18,7,19,5,21)(2,9,11,28,10,26,20,27)(3,13,16,17,15,31,14,30)(8,22,12,25,24,32,23,29);(3,14,15,16)(8,23,24,12)(13,30,31,17)(22,29,32,25);(2,10)(8,24)(9,26)(11,20)(12,23)(22,32)(25,29)(27,28)

Polycyclic presentation:

2^9|c^2=h,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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
maximal normal subgroup of	#72249829
maximal quotient of	#72257140

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368712);