#4380613

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2098 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)(5,6,8,10)(7,9,11,12)(13,14,16,18)(15,19,20,17)(22,26)(23,29)(28,34)(30,31);(13,14)(15,19)(16,18)(17,20)(21,22,25,26)(23,27,29,33)(24,28,32,34)(30,35,31,36);(5,6)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,20,19)(21,23)(22,27)(24,31)(25,29)(26,33)(28,36)(30,32)(34,35);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,27)(24,28)(25,26)(29,33)(30,35)(31,36)(32,34);(5,7,8,11)(6,9,10,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,32)(22,28,26,34)(23,30,29,31)(27,35,33,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4287809);