#1647707

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2014 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)(7,8)(9,10)(11,12)(13,14,16,18)(15,17,20,19)(21,22)(23,28)(24,32)(25,35)(26,36)(27,29)(30,31)(33,34);(6,10)(8,11)(13,14)(15,19)(16,18)(17,20)(21,23,27,24)(22,28,29,32)(25,33,36,30)(26,31,35,34);(5,7,9,12)(6,8,10,11)(13,14,16,18)(15,17,20,19)(21,24,27,23)(22,28,29,32)(25,33,36,30)(26,34,35,31);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,26)(23,30)(24,33)(27,36)(28,34)(29,35)(31,32);(5,8,9,11)(6,7,10,12)(13,15)(14,17)(16,20)(18,19)(21,26)(22,25)(23,31)(24,34)(27,35)(28,33)(29,36)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1554903);