#7969174

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	#41933	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4864 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)(3,4,6,7)(10,14)(15,16)(17,18,21,22)(19,25,26,23)(20,24,30,28)(27,32,29,31);(1,3)(2,4)(5,6)(7,8)(9,10,13,14)(11,15,12,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31);(1,4)(2,6)(3,8)(5,7)(9,11,13,12)(10,15,14,16)(18,22)(19,26)(24,28)(27,29);(1,3)(2,7)(4,8)(5,6)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32);(1,3)(2,4)(5,6)(7,8)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31);(1,4)(2,6)(3,8)(5,7)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7876370);