#7875522

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	#44138	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4742 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=ghi,f^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, 7782718);