#8195919

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5379 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=h,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284341, #72312299, #72312302, …
maximal normal subgroup of	#72284341, #72312299, #72312302, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103115);