#7654566

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×#3520	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7561762);