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H(5,𝔽5) #15691132

order3125
nilpotency class2
isoclinismisoclinic class of H(5,𝔽5)
automorphism?
homologyAbelianization (H1) = C54 · Schur multiplier (H2) = C55

Aliases

extraspecial5+1+4
HeisenbergH(5,𝔽5)

Subgroups

subgroupG/H
center, commutator, Frattini, socle, cosocleC5C54
LCSG ▷ C5 ▷ C1
UCSC1 ◁ C5 ◁ G
LECSG ▷ C5 ▷ C1

1433 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 125
transitive degreeμt(G) = 125

Polycyclic presentation:

5^5|b^a=be,d^a=de,c^b=ce

Conjugacy classes & character table

629 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#30626978, C5×H(5,𝔽5), #30627404, …
maximal normal subgroup of#30626978, C5×H(5,𝔽5), #30627404, …
maximal quotient ofC5×H(5,𝔽5), #30626982, H(3,𝔽5)2, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3125, 75);