#7968069

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gh,e^2=g,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, 7875265);