#7685358

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	#36066	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5250 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)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,29)(20,26)(21,27)(22,28)(23,25)(24,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,15,16,13)(17,19)(18,24)(20,31)(21,28)(22,32)(23,30)(25,29)(26,27);(1,2)(3,7)(4,5)(6,8)(9,12,14,11)(10,15,16,13)(18,25)(19,30)(20,28)(27,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,15)(10,12,16,11)(17,20,23,28)(18,26,25,22)(19,31,30,21)(24,27,29,32);(17,21,23,31)(18,27,25,32)(19,28,30,20)(22,24,26,29);(17,22,23,26)(18,28,25,20)(19,32,30,27)(21,29,31,24)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=ghi,e^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592554);