#1680629

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

3313 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,d^2=i,e^2=fgh,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	#72331994, #72332011, #72332051
maximal normal subgroup of	#72331994, #72332011, #72332051
maximal quotient of	#72333035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1587825);