#7737488

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	#44892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4921 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:

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

Polycyclic presentation:

2^9|a^2=h,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, 7644684);