#7671645

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5090 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578841);