#7631819

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	#36259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5010 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)(3,8,4,7)(9,10,13,16)(11,14,12,15)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,29,31);(1,3,6,4)(2,7,5,8)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,26)(21,29)(22,24)(25,28)(27,30)(31,32);(1,4,6,3)(2,8,5,7)(9,10)(11,14)(12,15)(13,16)(17,18,22,26)(19,20,24,23)(21,25,30,32)(27,28,29,31);(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,30)(31,32);(17,20)(18,24)(19,26)(21,28)(22,23)(25,29)(27,32)(30,31);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,15,16,14)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,f^2=h,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539015);