#7655452

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

6870 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)(5,6,11,13)(7,14,9,12)(8,17,19,10)(15,16,18,20);(1,2,3,4)(21,22)(23,28)(24,27)(25,29)(26,31)(30,34)(32,35)(33,36);(5,7,11,9)(6,12,13,14)(8,15,19,18)(10,16,17,20)(21,23)(22,26)(24,30)(25,31)(27,33)(28,29)(32,36)(34,35);(5,8)(6,10)(7,15)(9,18)(11,19)(12,16)(13,17)(14,20)(21,24,25,32)(22,27,29,35)(23,30,31,36)(26,33,28,34);(5,9,11,7)(6,12,13,14)(8,18,19,15)(10,16,17,20)(21,23)(22,28)(24,30)(25,31)(26,29)(27,34)(32,36)(33,35);(5,10)(6,8)(7,16)(9,20)(11,17)(12,15)(13,19)(14,18)(21,22)(23,26)(24,27)(25,29)(28,31)(30,33)(32,35)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=i,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, 7562648);