#8037025

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

7033 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7944221);