#6355464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355448
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60958	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306793, #72306795, #72306846
maximal normal subgroup of	#72306793, #72306795, #72306846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262660);