#1574559

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×#2408	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,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.

Relationships

maximal subgroup of	#72334129, #72334139, #72334146
maximal normal subgroup of	#72334129, #72334139, #72334146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1481755);