#7737489

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	#44895	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4835 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=ghi,e^2=gi,f^2=g,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, 7644685);