#6347372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268465, #72268468
maximal normal subgroup of	#72268465, #72268468

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254568);