#7685058

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	#45319	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4712 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592254);