#766890

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2482 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

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, 674086);