#7658643

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	C₂×#3516	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6290 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565839);