#7738211

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4694 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7645407);