#7849700

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

5042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7756896);