#7744971

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=gh,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, 7652167);