#10521299

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

5248 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28);(1,3,5,4)(2,6,7,8)(10,13)(14,15);(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,14,13,15)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28);(1,4,5,3)(2,6,7,8)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30);(1,2)(3,6)(4,8)(5,7)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428495);