#1581520

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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2993 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=hi,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	#72297723, #72297735, #72297751
maximal normal subgroup of	#72297723, #72297735, #72297751
maximal quotient of	#72298378

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1488716);