#7806585

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	#34977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4814 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,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, 7713781);