#92525

order	500
isoclinism	isoclinic class of #6953
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

derived, Sylow

H(3,𝔽₅) ⋊ C₄

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂×C₅	C₅⋊D₅
commutator, hypocenter	H(3,𝔽₅)	C₄
Fitting, prime radical, cosocle	C₂×H(3,𝔽₅)	C₂
LCS	G ▷ H(3,𝔽₅)
UCS	C₁ ◁ C₂×C₅

45 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 3 generators on 29 points:

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

Polycyclic presentation:

2^2,5^3|a^2=b,c^a=c^4,d^a=d^4,d^c=de

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#14522305, #14522306, #14522307
maximal subgroup of	#11758647, #11758649, #11758657, …
maximal normal subgroup of	#11758647, #11758649, #11758657, …
maximal quotient of	#11758635, C₂×#92525, C₃×#92525, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 11);