#1262470

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2605 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1169666);