#6496270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470237
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	C₂⁴	C₂⁵
prime radical	#62781	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3069 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317700, #72317718, #72317780, …
maximal normal subgroup of	#72317700, #72317718, #72317780, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6403466);