#6353458

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353301
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#63025	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2643 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283267
maximal normal subgroup of	#72283267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260654);