#7672301

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579497);