#7820139

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

4731 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=gh,f^2=g,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, 7727335);