#7857529

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4784 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)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18,23,24)(19,22,27,32)(20,25,29,31)(21,30,28,26);(9,10)(11,14)(12,13)(15,16)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,26,31,30);(2,5)(6,7)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(20,29)(26,30);(1,2)(3,6)(4,5)(7,8)(17,20)(18,25)(19,28)(21,27)(22,26)(23,29)(24,31)(30,32);(1,3,4,8)(2,6,5,7)(17,21,23,28)(18,26,24,30)(19,20,27,29)(22,31,32,25);(17,22)(18,27)(19,24)(20,26)(21,25)(23,32)(28,31)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7764725);