#7631789

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	#36020	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5010 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,f^2=h,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, 7538985);