#7643642

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
prime radical	C₂²×#538	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550838);