#7737555

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	#59213	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4719 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644751);