#7746085

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	#35844	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4792 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7653281);