#6366505

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366374
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1592 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339681, #72339713, #72339747
maximal normal subgroup of	#72339681, #72339713, #72339747

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273701);