#7776124

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4611 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:

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

Polycyclic presentation:

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

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, 7683320);