#8107217

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
prime radical	#36123	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4949 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014413);