#7849682

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	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4852 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7756878);