#7631661

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	#35000	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5468 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538857);