#7685524

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	#59024	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4856 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18,23,25)(19,22,28,24)(20,26,29,31)(21,27,30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,23,28)(18,24,25,22)(20,21,29,30)(26,32,31,27);(2,7)(6,8)(18,25)(19,28)(20,29)(27,32);(1,2)(3,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(17,20,23,29)(18,26,25,31)(19,21,28,30)(22,27,24,32);(1,4,5,3)(2,8,7,6)(17,21)(18,27)(19,29)(20,28)(22,31)(23,30)(24,26)(25,32);(17,22,23,24)(18,28,25,19)(20,27,29,32)(21,26,30,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=hi,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, 7592720);