#7875307

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	#44140	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=hi,e^2=g,f^2=gh,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, 7782503);