#7711152

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5362 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:

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

Polycyclic presentation:

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

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, 7618348);