#7710420

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

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

6464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(24,31)(28,33)(29,35)(34,36);(1,2,3,4)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(6,7)(11,13)(12,14)(18,19)(22,26)(27,32)(28,33)(34,36);(5,7,10,6)(8,13,16,11)(9,14,17,12)(15,19,20,18)(21,23,25,30)(22,27,26,32)(24,29,31,35)(28,34,33,36);(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,22)(23,27)(24,28)(25,26)(29,34)(30,32)(31,33)(35,36);(5,9,10,17)(6,12,7,14)(8,15,16,20)(11,18,13,19)(21,24)(22,28)(23,29)(25,31)(26,33)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=hi,e^2=h,f^2=h,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, 7617616);