#7630845

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	#36018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538041);