#7759723

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	#59808	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4909 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)(17,18,22,23)(19,20,25,28)(21,24,29,30)(26,27,31,32);(1,2)(3,4)(5,6)(7,8)(9,10,13,14)(11,15,12,16)(17,19,22,25)(18,20,23,28)(21,26,29,31)(24,27,30,32);(2,6)(4,7)(9,11,13,12)(10,15,14,16)(18,23)(19,25)(21,29)(27,32);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,15,14,16)(17,19)(18,20)(21,26)(22,25)(23,28)(24,27)(29,31)(30,32);(1,3)(2,7)(4,6)(5,8)(9,12,13,11)(10,16,14,15)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(1,4,5,7)(2,3,6,8)(17,21,22,29)(18,24,23,30)(19,26,25,31)(20,27,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7666919);